definition of geometry
DESCRIPTION
Definition of Geometry. The branch of mathematics concerned with properties of and relations between points, lines, planes and figures. Origins Of Geometry. - PowerPoint PPT PresentationTRANSCRIPT
Definition of GeometryDefinition of Geometry
The branch of mathematics concerned The branch of mathematics concerned with properties of and relations with properties of and relations between points, lines, planes and between points, lines, planes and figures.figures.
Origins Of GeometryOrigins Of GeometryThe earliest records of Geometry can be The earliest records of Geometry can be
traced to ancient Egypt and the Indus traced to ancient Egypt and the Indus Valley from around 3000 BC. Early Valley from around 3000 BC. Early Geometry was a collection of observed Geometry was a collection of observed principles concerning lengths, angles, principles concerning lengths, angles, area, and volumes. These principles were area, and volumes. These principles were developed to meet practical needs in developed to meet practical needs in construction, astronomy, and other needs.construction, astronomy, and other needs.
Origins of GeometryOrigins of GeometryEuclid, a Greek mathematician, wrote Euclid, a Greek mathematician, wrote The The
Elements of GeometryElements of Geometry. It is considered . It is considered one of the most important early texts on one of the most important early texts on Geometry.Geometry.
He presented geometry in a practical form He presented geometry in a practical form known as Euclidean geometry. Euclid was known as Euclidean geometry. Euclid was not the first elementary Geometry not the first elementary Geometry textbook, but the others fell into disuse textbook, but the others fell into disuse and were lost.and were lost.
Chapter 1Chapter 1
Basics of GeometryBasics of Geometry
1.1 Patterns and Inductive 1.1 Patterns and Inductive ReasoningReasoning
Vocabulary:Vocabulary:A A conjectureconjecture is an unproven statement is an unproven statement
that is based on observations.that is based on observations.Example:Example:3 + 4 + 5 = 4 x 3 6 + 7 + 8 = 7 x 33 + 4 + 5 = 4 x 3 6 + 7 + 8 = 7 x 34 + 5 + 6 = 5 x 3 7 + 8 + 9 = 8 x 34 + 5 + 6 = 5 x 3 7 + 8 + 9 = 8 x 35 + 6 + 7 = 6 x 3 8 + 9 + 10 = 9 x 35 + 6 + 7 = 6 x 3 8 + 9 + 10 = 9 x 3Conjecture: The sum of any three Conjecture: The sum of any three
consecutive integers is consecutive integers is 3 times the3 times the middle number. middle number.
Example:Example:
Conjecture: The sum of any two odd Conjecture: The sum of any two odd numbers is ____.numbers is ____.
1 + 1 = 2 1 + 3 = 4 3 + 5 = 81 + 1 = 2 1 + 3 = 4 3 + 5 = 8
7 + 11 = 18 13 + 19 = 327 + 11 = 18 13 + 19 = 32
Inductive reasoningInductive reasoning is a process that is a process that includes looking for patterns and includes looking for patterns and making conjectures. Prediction or making conjectures. Prediction or conclusion based on observation of a conclusion based on observation of a pattern.pattern.
Example—expecting a traffic light to Example—expecting a traffic light to stay green for 40 seconds because stay green for 40 seconds because you have seen it stay green for 40 you have seen it stay green for 40 seconds many times before.seconds many times before.
Example—When you see the numbers Example—When you see the numbers 2, 4, 6, 8, and 10, and you expect the 2, 4, 6, 8, and 10, and you expect the next number to be 12.next number to be 12.
Predict the next number.Predict the next number.
2, 6, 18, 54, …2, 6, 18, 54, …
A A counterexamplecounterexample is an example that is an example that shows a conjecture is false.shows a conjecture is false.
Example: Show the conjecture is false Example: Show the conjecture is false by finding a counterexample:by finding a counterexample:
The difference of two positive The difference of two positive numbers is always positive.numbers is always positive.
Answer: 3 – 9 = -6Answer: 3 – 9 = -6
Describe pattern and predict next Describe pattern and predict next numbernumber
1. 128, 64, 32, 16, …1. 128, 64, 32, 16, …
2. 5, 4, 2, -1, …2. 5, 4, 2, -1, …
ExampleExample Given the pattern __, -6, 12, __, 48, … Given the pattern __, -6, 12, __, 48, …
answer the following exercises:answer the following exercises: a. Fill in the missing numbers.a. Fill in the missing numbers.
b. Determine the next two numbers in b. Determine the next two numbers in
this sequence.this sequence.
c. Describe how you determined what c. Describe how you determined what
numbers completed the sequence.numbers completed the sequence.
d. Are there any other numbers that d. Are there any other numbers that
would complete this sequence?would complete this sequence?
Answer to previous question.Answer to previous question. A) 3, -6, 12, -24, 48A) 3, -6, 12, -24, 48 B) the next two numbers are -96 and B) the next two numbers are -96 and
192192 D) -24, -6, 12, 30, 48D) -24, -6, 12, 30, 48
next two numbers: 66 and 84next two numbers: 66 and 84
Complete the ConjectureComplete the Conjecture
The sum of the first The sum of the first nn even positive even positive integers is _____?integers is _____?
11stst even integer: 2 = 1(2) even integer: 2 = 1(2)
Sum of 1Sum of 1stst two even two even
pos. integers: 2 + 4 = 6 = 2(3)pos. integers: 2 + 4 = 6 = 2(3)
Sum of 1Sum of 1stst three…: 2 + 4 + 6 = 12=3(4) three…: 2 + 4 + 6 = 12=3(4)
Sum of 1Sum of 1stst four…: 2 + 4 + 6 + 8 = 20= 4(5) four…: 2 + 4 + 6 + 8 = 20= 4(5)
Sum of 1Sum of 1stst n even pos. Int. is: n(n + 1) n even pos. Int. is: n(n + 1)
CounterexampleCounterexample
Show the conjecture is false by Show the conjecture is false by finding a counterexample.finding a counterexample.
Conjecture: If the difference of two Conjecture: If the difference of two numbers is odd, then the greater of numbers is odd, then the greater of the two numbers must also be odd.the two numbers must also be odd.
Finding the nth termFinding the nth term
http://www.math-magic.com/sequences/nth_term.htm
1.2 Points, Lines, Planes1.2 Points, Lines, Planes
The three The three undefined termsundefined terms are point, line, are point, line, and plane.and plane.
A A pointpoint has no dimension. has no dimension.
A A pointpoint is usually named by a capital is usually named by a capital letter.letter.
AA ٭٭ All geometric figures consist of points.All geometric figures consist of points.
LinesLines
LinesLines extend indefinitely and have no extend indefinitely and have no thickness or width.thickness or width.
LinesLines are usually named by lower case are usually named by lower case script letters or by writing capital letters for script letters or by writing capital letters for 2 points on the line, with a double arrow 2 points on the line, with a double arrow over the pair of letters.over the pair of letters.
l line ll line l
A B C AB BC AC A B C AB BC AC
PlanePlane A A planeplane extends in two dimensions. It extends in two dimensions. It
is usually represented by a shape that is usually represented by a shape that looks like a tabletop or wall.looks like a tabletop or wall.
A A planeplane is a flat surface that extends is a flat surface that extends indefinitely in all directions.indefinitely in all directions.
A A planeplane can be named by a capital can be named by a capital script letter or by three noncollinear script letter or by three noncollinear points in the plane. plane ABC or points in the plane. plane ABC or plane plane RR
http://www.mathopenref.com/plane.html
SpaceSpace-Three dimensional set of all -Three dimensional set of all points.points.
Collinear pointsCollinear points are points that lie on are points that lie on the same line.the same line.
Coplanar pointsCoplanar points are points that lie on are points that lie on the same plane. the same plane.
A
Line Segment, EndpointLine Segment, Endpoint
A A line segmentline segment is part of a line that is part of a line that consists of two points, called consists of two points, called endpoints, and all points on the line endpoints, and all points on the line between the endpoints. Name by between the endpoints. Name by using endpoints.using endpoints.
B ABB AB
A
Ray, Initial pointRay, Initial point
A A rayray is part of a line that consists of is part of a line that consists of a point, called an initial point, and all a point, called an initial point, and all points on the line that extend in one points on the line that extend in one direction. Name by using the direction. Name by using the endpoint first, then any point of ray.endpoint first, then any point of ray.
A B AB
Opposite RaysOpposite Rays If C is between A and B on AB, then If C is between A and B on AB, then
CA and CB are opposite rays.CA and CB are opposite rays.
B C A
Two or more geometric figures Two or more geometric figures intersectintersect if if they have one or more points in common.they have one or more points in common.
The intersection of two or more geometric figures is the set of points that the figures have in common.
HomeworkHomework
Pages 6 – 9 #’s 12 – 15, 16 – 22 Pages 6 – 9 #’s 12 – 15, 16 – 22 evens, 25 30, 47, 48evens, 25 30, 47, 48
Pages 13 – 16 #’s 10 – 16 evens, 25 Pages 13 – 16 #’s 10 – 16 evens, 25 – 31 odds, 37, 44 - 47– 31 odds, 37, 44 - 47
1.3 Segments and their Measures1.3 Segments and their Measures PostulatesPostulates are rules that are are rules that are
accepted without proof. Postulates accepted without proof. Postulates are also called axioms.are also called axioms.
Ex: A line contains at least two points.Ex: A line contains at least two points. A A coordinatecoordinate is a real number that is a real number that
corresponds to a point on a line.corresponds to a point on a line. The The distancedistance between two points on between two points on
a line is the absolute value of the a line is the absolute value of the difference between the coordinates difference between the coordinates of the points.of the points.
The The lengthlength of a segment is the of a segment is the distance between the endpoints.distance between the endpoints.
When three points lie on a line, you When three points lie on a line, you can say that one of them is can say that one of them is betweenbetween the other two.the other two.
The The Distance FormulaDistance Formula is a formula for is a formula for finding the distance between two finding the distance between two points in a coordinate plane.points in a coordinate plane.
Congruent segmentsCongruent segments are segments are segments that have the same length.that have the same length.
Postulate 1: Ruler PostulatePostulate 1: Ruler Postulate
The points on a line can be matched The points on a line can be matched one to one with real numbers. The one to one with real numbers. The real number that corresponds to a real number that corresponds to a point is the point is the coordinatecoordinate of the point. of the point.
The distance between points A and The distance between points A and B, written AB, is the absolute value of B, written AB, is the absolute value of the difference between the the difference between the coordinates of A and B. AB is also coordinates of A and B. AB is also called the called the lengthlength of AB. of AB.
Finding distance between two Finding distance between two pointspoints
AB = AB =
Find the distance between the pointsFind the distance between the points 12) E and A12) E and A 13) F and B13) F and B 14) E and D14) E and D 15) C and B15) C and B 16) F and A16) F and A
Distance Formula:Distance Formula: Given the two Given the two points (points (xx11, , yy11) and () and (xx22, , yy22), the ), the distance between these points is distance between these points is given by the formula:given by the formula:
d = d = )()( 1212 yyxx 2 2
ExampleExample
B (3, 4)
C(4, -3)
D(-3,-2)
A(-1,2)
Find AB, BC, CD, and AD.
Postulate 2: Segment Addition Postulate 2: Segment Addition PostulatePostulate
If B is between A and C, thenIf B is between A and C, then
AB + BC = AC. AB + BC = AC. If AB + BC = AC, then B is between A If AB + BC = AC, then B is between A
and C.and C.
A lies between C and T. Find CT if CA A lies between C and T. Find CT if CA is 5 and AT is 8.is 5 and AT is 8.
Find AC if CT is 20 and AT is 8.Find AC if CT is 20 and AT is 8. See Notetaking guide book.See Notetaking guide book.
1.4 Angles and Their Measures1.4 Angles and Their Measures
An An angleangle consists of two noncollinear consists of two noncollinear rays that have the same initial point.rays that have the same initial point.
A
B
C
< ABC with sides
BA and BC
The initial point of the rays is the vertex of the angle. The vertex is point B.
Name the angle:
<ABC or
<CBA or
<B or
<1
1
Congruent angles are angles that Congruent angles are angles that have the same measure.have the same measure.
Congruent anglesCongruent angles
A B
m<A = m<B
Measure of an angleMeasure of an angle
In <AOB, ray OA and ray OB can be In <AOB, ray OA and ray OB can be matched one to one with the real matched one to one with the real numbers from 0 to 180.numbers from 0 to 180.
The measure of <AOB is equal to the The measure of <AOB is equal to the absolute value of the difference absolute value of the difference between the real numbers forbetween the real numbers for
ray OA and ray OB.ray OA and ray OB.
Acute AngleAcute Angle
An acute angle is an angle that An acute angle is an angle that measures between 0° and 90°.measures between 0° and 90°.
http://optics.org/cws/article/research/23663
AnglesAngles
An angle separates An angle separates a plane into three a plane into three parts: the parts: the interiorinterior,, the the exteriorexterior, and , and the the angle itself.angle itself.
exterior interior
Right AngleRight Angle
A right angle is an angle that A right angle is an angle that measures 90°.measures 90°.
Obtuse AngleObtuse Angle
An obtuse angle is an angle that An obtuse angle is an angle that measures between 90° and 180°.measures between 90° and 180°.
Straight AngleStraight Angle
A straight angle is an angle that A straight angle is an angle that measures 180º.measures 180º.
Adjacent AnglesAdjacent Angles
Two angles are adjacent if they have Two angles are adjacent if they have a common vertex and side, but have a common vertex and side, but have no common interior points.no common interior points.
A B CB C
DD
A
<ABD AND <DBC
are adjacent
<ABD AND <DBC
are adjacent
Adjacent AnglesAdjacent Angles
1
2
3
4
Adjacent Angles: <1 and <2; <2 and <3;
<3 and <4; <4 and <1
NOTE: Not adjacent
<1 and <3, <4 and <2
Name the angles in the figureName the angles in the figure
J ML
K
Protractor PostulateProtractor Postulate For every angle there is a unique real For every angle there is a unique real
number r, called its degree measure, number r, called its degree measure, such that 0 < r < 180.such that 0 < r < 180.
*ILLUSTRATION ON NEXT SLIDE
A
O
BA
Angle Addition PostulateAngle Addition Postulate
If If P P is in the interior is in the interior of <RST, thenof <RST, then
m<RSP + m<PST =m<RSP + m<PST =
m<RSTm<RST
R
P
T
S
Measure the angle. Then classify the
angle as acute, right, obtuse, or straight.
a. <AFD
b. <AFE
c. <BFD
d. <BFC
HomeworkHomework
Pages 21 – 24 #’s 20, 24 – 30 evens, Pages 21 – 24 #’s 20, 24 – 30 evens, 31, 33, 35, 48, 55, 5631, 33, 35, 48, 55, 56
Pages 29 – 32 #’s 14 – 34 evens, 68, Pages 29 – 32 #’s 14 – 34 evens, 68, 70 - 7370 - 73
1.5 Segment and Angle Bisector1.5 Segment and Angle Bisector
A midpoint is the point that divides, A midpoint is the point that divides, or bisects, a segment into two or bisects, a segment into two congruent segmentscongruent segments..
The midpoint M of PQ is the point The midpoint M of PQ is the point between P and Q such that PM = MQ.between P and Q such that PM = MQ.
PM Q
Midpoint of SegmentMidpoint of Segment
If B is the midpoint of segment AC If B is the midpoint of segment AC and AB = 2x + 8 and BC = 4x – 2 and AB = 2x + 8 and BC = 4x – 2
find AB, BC, and AC.find AB, BC, and AC.
ConstructionConstruction You can use a compass and a You can use a compass and a
straightedge to find the midpoint of a straightedge to find the midpoint of a segment.segment.
A construction is a geometric A construction is a geometric drawing that is created using a drawing that is created using a limited set of tools, usually a limited set of tools, usually a compass and a straightedge.compass and a straightedge.
A compass is a tool used to draw A compass is a tool used to draw arcs.arcs.
A straightedge used to draw A straightedge used to draw segments.segments.
Show a construction of a midpoint.Show a construction of a midpoint.
Midpoint ProblemMidpoint Problem
Example:Example: If the coordinate of H is -5 and the If the coordinate of H is -5 and the
coordinate of J is 4, what is the coordinate of J is 4, what is the coordinate of the midpoint of HJ?coordinate of the midpoint of HJ?
Midpoint Formula for Coordinate Midpoint Formula for Coordinate PlanePlane
Find the coordinates of the midpoint of AB with endpoints
A(-3, -4) and B(5, 5)
Midpoint ProblemMidpoint Problem
The midpoint of JK is M(1, 4). One The midpoint of JK is M(1, 4). One endpoint is J(-3, 2). Find the endpoint is J(-3, 2). Find the coordinates of the other endpoint.coordinates of the other endpoint.
The midpoint of PQ is M(5, 3). One The midpoint of PQ is M(5, 3). One endpoint is P(-5, 12). Find the endpoint is P(-5, 12). Find the coordinates of the other endpoint.coordinates of the other endpoint.
Bisect a segmentBisect a segment
To bisect a segment or an angle To bisect a segment or an angle means to divide it into two congruent means to divide it into two congruent parts.parts.
l
AB
Line l bisects segment AB.
Segment bisectorSegment bisector
Construct a segment bisector using a Construct a segment bisector using a compass and protractor.compass and protractor.
Angle BisectorAngle Bisector
An angle bisector is a ray that An angle bisector is a ray that divides an angle into two adjacent divides an angle into two adjacent angles that are congruent.angles that are congruent.
m<ABC = m<CBDm<ABC = m<CBD A
B
C
D
Bisect an angleBisect an angle RT is the angle bisector of <QRS. RT is the angle bisector of <QRS.
Given that m<QRS = 42º, what are Given that m<QRS = 42º, what are the measures of <QRT and <TRS?the measures of <QRT and <TRS?
Q
T
S
R
KM bisects <JKL.KM bisects <JKL.
m<JKM = 2x + 7m<JKM = 2x + 7 m<MKL = 4x – 41m<MKL = 4x – 41 Find m<JKM and m<MKL.Find m<JKM and m<MKL.
J
KM
L
Angle BisectorAngle Bisector
BD is the angle bisector of <ABC. BD is the angle bisector of <ABC. Find the m<ABD and m<DBC.Find the m<ABD and m<DBC.
A
B
C
D
10x - 256x + 15
1.6 Angle Pair Relationships1.6 Angle Pair Relationships
Vertical angles consist of two angles Vertical angles consist of two angles whose sides form two pairs of whose sides form two pairs of opposite rays. Vertical angles are opposite rays. Vertical angles are congruent.congruent.
<1 and <3 and <2 and <4 are <1 and <3 and <2 and <4 are vertical angles.vertical angles.
12
3
4
Linear PairLinear Pair
A linear pair consists of two adjacent A linear pair consists of two adjacent angles whose noncommon sides are angles whose noncommon sides are opposite rays. Linear pairs of angles opposite rays. Linear pairs of angles are supplementary. (Sum of angles is are supplementary. (Sum of angles is 180°.)180°.)
<1 and <2, <2 and <3, <3 and <4,<1 and <2, <2 and <3, <3 and <4,
and <4 and <1 are linear pairs of and <4 and <1 are linear pairs of angles.angles.
12
34
ExampleExample A) Are <1 and <2 a linear pair?A) Are <1 and <2 a linear pair? B) Are <4 and <5 a linear pair?B) Are <4 and <5 a linear pair? C) Are <5 and <3 vertical angles?C) Are <5 and <3 vertical angles? D) Are <1 and <3 vertical angles?D) Are <1 and <3 vertical angles?
23
45
1
ExampleExample Solve for x and y. Then find the Solve for x and y. Then find the
angle measures.angle measures.
4x + 15
5x + 30
3y + 15
3y - 15
ExampleExample A) Name one pair of vertical angles A) Name one pair of vertical angles
and one pair of angles that form a and one pair of angles that form a linear pair.linear pair.
B) What is the measure of <GHI in B) What is the measure of <GHI in the figure above?the figure above?
I
H
G K
J
5x + 30
2x - 4
Complementary AnglesComplementary Angles
Complementary angles are two Complementary angles are two angles whose measures have the angles whose measures have the sum 90°. sum 90°.
Complement: The sum of the Complement: The sum of the measures of an angle and its measures of an angle and its complement is 90°.complement is 90°.
2070
or
Supplementary AnglesSupplementary Angles Supplementary angles are two Supplementary angles are two
angles whose measures have the angles whose measures have the sum 180°.sum 180°.
Supplement: The sum of the Supplement: The sum of the measures of an angle and its measures of an angle and its supplement is 180°.supplement is 180°.
120
60
or
ExamplesExamples 1. Given that <A is a complement of 1. Given that <A is a complement of
<C and m<A = 47°, find m<C.<C and m<A = 47°, find m<C. 2. Given that <P is a supplement of 2. Given that <P is a supplement of
<R and m<R = 36°, find m<P.<R and m<R = 36°, find m<P. 3. <W and <Z are complementary. 3. <W and <Z are complementary.
The measure of <Z is 5 times the The measure of <Z is 5 times the measure of <W. Find m<W.measure of <W. Find m<W.
4. <T and <S are supplementary. 4. <T and <S are supplementary. The measure of <T is half the The measure of <T is half the measure of <S. Find m<S.measure of <S. Find m<S.
ExamplesExamples 5. When two lines intersect, the 5. When two lines intersect, the
measure of one of the angles they measure of one of the angles they form is 20° less than three times the form is 20° less than three times the measure of one of the other angles measure of one of the other angles formed. What are the measures of formed. What are the measures of all four angles formed by the lines?all four angles formed by the lines?
Homework
Pages 38 – 40 #’s 18, 24 – 30 evens, 38 – 42 evens, 44 - 49
Pages 47 – 50 #’s 8 – 26 evens, 28 – 36 evens, 43, 44, 46 – 52 evens
Combinations and PermutationsCombinations and Permutations
The rest of the slides in this chapter The rest of the slides in this chapter come from these two sites. GLE’scome from these two sites. GLE’s
http://www.mathsisfun.com/combinatorics/combinations-permutations.html
http://www.glencoe.com/sec/math/prealg/prealg03/extra_examples/chapter12/lesson12_7.pdf
Chapter 1 TestChapter 1 Test
Pages 60 – 63 Chapter Review and Pages 60 – 63 Chapter Review and Chapter TestChapter Test
Combinations and PermutationsCombinations and Permutations What’s the Difference?What’s the Difference?
Combination—order doesn’t matter.Combination—order doesn’t matter. Example: My fruit salad is a Example: My fruit salad is a
combination of apples, grapes and combination of apples, grapes and bananas. We don’t care what order bananas. We don’t care what order the fruits are in.the fruits are in.
Permutation—order does matter. To Permutation—order does matter. To help you to remember, thinkhelp you to remember, think
“ “Permutation …Position”Permutation …Position”
Example: The combination to the Example: The combination to the safe was 472. We do care about the safe was 472. We do care about the order. “724” would not work, nor order. “724” would not work, nor would “247”. It has to be exactly 4-would “247”. It has to be exactly 4-7-2.7-2.
PermutationsPermutations There are two types of permutations:There are two types of permutations:
1. Repetition is Allowed: such as 1. Repetition is Allowed: such as
the lock on previous slide. It the lock on previous slide. It
could be “333”.could be “333”.
2. No Repetition: for example the 2. No Repetition: for example the
first three people in a running first three people in a running
race. You can’t be 1race. You can’t be 1stst and 2 and 2ndnd..
Permutations with RepetitionPermutations with Repetition
Calculate-- If you have n things to choose Calculate-- If you have n things to choose from, and you choose r of them, then the from, and you choose r of them, then the permutations are:permutations are:
r r n x n x … (r times) = n n x n x … (r times) = n
where n is the number of things to choose where n is the number of things to choose from, and you choose r of them (Repetition from, and you choose r of them (Repetition allowed, order matters)allowed, order matters)
Example (Permutations with Example (Permutations with repetition)repetition)
Combination Lock:Combination Lock:
There are 10 numbers to choose from There are 10 numbers to choose from (0, 1, …, 9) and you choose 3 of (0, 1, …, 9) and you choose 3 of them:them:
10 x 10 x … (3 times) = 10 = 100010 x 10 x … (3 times) = 10 = 1000 permutationspermutations
Ex: Perm. With RepetitionEx: Perm. With Repetition Police use photographs of various Police use photographs of various
facial features to help witnesses facial features to help witnesses identify suspects. One basic identify suspects. One basic information kit contains 195 information kit contains 195 hairlines, 99 eyes and eyebrows, 89 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and noses, 105 mouths, and 74 chins and cheeks. cheeks.
How many different faces can be How many different faces can be produced?produced?
AnswerAnswer
Number of faces:Number of faces:
195 x 99 x 89 x 105 x 74 =195 x 99 x 89 x 105 x 74 =
Hairlines x eyes x noses x mouths x Hairlines x eyes x noses x mouths x chinschins
13,349,986,65013,349,986,650
ExampleExample The standard configuration for a New The standard configuration for a New
York license plate is 3 digits followed York license plate is 3 digits followed by 3 letters. How many different by 3 letters. How many different license plates are possible if digits license plates are possible if digits and letters can be repeated?and letters can be repeated?
234 ABC
New York
AnswerAnswer
There are 10 choices for each digit There are 10 choices for each digit and 26 choices for each letter.and 26 choices for each letter.
Number of plates:Number of plates:
10 x 10 x 10 x 26 x 26 x 26 =10 x 10 x 10 x 26 x 26 x 26 =
17,576,00017,576,000
Permutations without RepetitionPermutations without Repetition
In this case, you have to reduce the In this case, you have to reduce the number of available choices each number of available choices each time.time.
Example w/o RepetitionExample w/o Repetition For example, what order could 16 For example, what order could 16
pool balls be in?pool balls be in?
(After choosing, say, number “14” (After choosing, say, number “14” you can’t choose it again.you can’t choose it again.
11stst choice: 16 possibilities, choice: 16 possibilities,
22ndnd choice: 15 poss., choice: 15 poss.,
Then 14, 13, etc. Then 14, 13, etc.
16 x 15 x 14 x …=20,922,789,888,00016 x 15 x 14 x …=20,922,789,888,000
continuedcontinued Maybe you don’t want to choose Maybe you don’t want to choose
them all, just 3 of them, so that them all, just 3 of them, so that would be only:would be only:
16 x 15 x 14 = 336016 x 15 x 14 = 3360
In other words, there are 3,360 In other words, there are 3,360 different ways that 3 pool balls could different ways that 3 pool balls could be selected out of 16 balls.be selected out of 16 balls.
Factorial FunctionFactorial Function
We can use the factorial symbol We can use the factorial symbol
( ! ) to help us write these functions ( ! ) to help us write these functions Mathematically.Mathematically.
4! = 4 x 3 x 2 x 1 = 244! = 4 x 3 x 2 x 1 = 24
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 =
50405040
Select all billiard balls:Select all billiard balls:
16! = 20,922,789,888,00016! = 20,922,789,888,000
But if you wanted to select just 3:But if you wanted to select just 3:
16! 16! = 16 x 15 x 14= 16 x 15 x 14
(16 – 3)!(16 – 3)!
= 3360= 3360
FormulaFormula
P(n, r) = P = P = P(n, r) = P = P = n! n!
(n – r)!(n – r)!
Where n is the number of things Where n is the number of things choose from, and you choose r of choose from, and you choose r of them (No repetition, order matters)them (No repetition, order matters)
n
r n r
Example: Perm. Example: Perm. No repetition, Order mattersNo repetition, Order matters
How many ways can 1How many ways can 1stst and 2 and 2ndnd place place be awarded to 10 people?be awarded to 10 people?
10!10!
(10 – 2)!(10 – 2)!
= 90= 90
Example: Perm. no repetitionExample: Perm. no repetition
The standard configuration for a New The standard configuration for a New York license plate is 3 digits followed York license plate is 3 digits followed by 3 letters.by 3 letters.
How many different license plates How many different license plates are possible if digits and letters are possible if digits and letters cannot be repeated?cannot be repeated?
10!10! X X 26!26! = 11,232,000 = 11,232,000
(10 – 3)! (26 – 3)!(10 – 3)! (26 – 3)!
Example: PermutationsExample: Permutations Twelve skiers are competing in the Twelve skiers are competing in the
final round of the Olympic freestyle final round of the Olympic freestyle skiing aerial competition.skiing aerial competition.
a) In how many different ways can a) In how many different ways can the skiers finish the competition? the skiers finish the competition? (Assume there are no ties.)(Assume there are no ties.)
b) In how many different ways can 3 b) In how many different ways can 3 of the skiers finish 1of the skiers finish 1stst, 2, 2ndnd, and 3, and 3rdrd to to win the gold, silver, and bronze win the gold, silver, and bronze metals?metals?
AnswersAnswers A) There are 12! Different ways that A) There are 12! Different ways that
the skiers can finish the competition. the skiers can finish the competition. = 479,001,600= 479,001,600
B) Any of the 12 skiers can finish 1B) Any of the 12 skiers can finish 1stst, , then any of the remaining 11 skiers then any of the remaining 11 skiers can finish 2can finish 2ndnd, and finally any of the , and finally any of the remaining 10 skiers can finish 3remaining 10 skiers can finish 3rdrd..
12!12! = 1320 = 1320
(12 – 3)!(12 – 3)!
Perm. problemPerm. problem You are considering 10 different You are considering 10 different
colleges. Before you decide to apply colleges. Before you decide to apply to the colleges, you want to visit to the colleges, you want to visit some or all of them. In how many some or all of them. In how many orders can you visit (a) 6 of the orders can you visit (a) 6 of the colleges and (b) all 10 colleges?colleges and (b) all 10 colleges?
AnswerAnswer
A) P = 10!/(10 – 6)! = 151,200A) P = 10!/(10 – 6)! = 151,200
B) P =10!/(10 – 10)! =3,628,800B) P =10!/(10 – 10)! =3,628,800
10 6
10 10
Note: 0! = 1
Permutation ExamplePermutation Example
How many ways can gold, silver, and How many ways can gold, silver, and bronze medals be awarded for a race bronze medals be awarded for a race run by 8 people?run by 8 people?
AnswerAnswer
P(8, 3) =P(8, 3) =
8!8! = 8 x 7 x 6 = 8 x 7 x 6 (8 – 3)! (8 choices for gold,(8 – 3)! (8 choices for gold, 7 choices for silver,7 choices for silver, 6 choices for bronze)6 choices for bronze)There are 336 possible ways to award There are 336 possible ways to award
the medals.the medals.
Example: PermutationExample: Permutation
How many five-digit zip codes can be How many five-digit zip codes can be made where all digits are unique? made where all digits are unique? The possible digits are the numbers 0 The possible digits are the numbers 0 – 9.– 9.
answeranswer
P(10, 5) = 10 x 9 x 8 x 7 x 6P(10, 5) = 10 x 9 x 8 x 7 x 6
10 choices for 110 choices for 1stst digit digit
9 choices for 29 choices for 2ndnd digit digit
8 choices for 38 choices for 3rdrd digit digit
7 choices for 47 choices for 4thth digit digit
6 choices for 36 choices for 3rdrd digit digit
= 30,240= 30,240
Perm. problemPerm. problem
A classroom has 24 seats and 24 A classroom has 24 seats and 24 students. Assuming the seats are students. Assuming the seats are not moved, how many different not moved, how many different seating arrangements are possible?seating arrangements are possible?
6.20 x 10^6.20 x 10^23
COMBINATIONSCOMBINATIONS
There are two types of There are two types of combinations (remember the order combinations (remember the order does not matter now):does not matter now):
1. Repetition is Allowed: such as 1. Repetition is Allowed: such as
coins in your pocket (5, 5, 5, 10, coins in your pocket (5, 5, 5, 10,
10)10)
2.2. No Repetition: such as lottery No Repetition: such as lottery
numbers (2, 14, 15, 27, 30, 33)numbers (2, 14, 15, 27, 30, 33)
Combinations without RepetitionCombinations without Repetition
This is how the lotteries work. The This is how the lotteries work. The numbers drawn one at a time, and if numbers drawn one at a time, and if you have the lucky numbers (no you have the lucky numbers (no matter what order) you win!matter what order) you win!
Order in not important.Order in not important.
CombinationsCombinations
Order does Order doesn’tOrder does Order doesn’t
matter mattermatter matter
1 2 3 1 2 3
1 3 21 3 2
2 1 3 1 2 32 1 3 1 2 3
2 3 12 3 1
3 1 23 1 2
3 2 13 2 16 diff combinations
One combination
Combination FormulaCombination Formula
C(n, r) = C = C = C(n, r) = C = C =
= = n!n!
r!(n – r)!r!(n – r)!
n
r n r
n
r
ExampleExample
A standard deck of 52 playing cards A standard deck of 52 playing cards has 4 suits with 13 different cards in has 4 suits with 13 different cards in each suit as shown. If the order in each suit as shown. If the order in which the cards are dealt is not which the cards are dealt is not important, how many different 5-card important, how many different 5-card hands are possible?hands are possible?
AnswerAnswer
The number of ways to choose 5 The number of ways to choose 5 cards from a deck of 52 cards is:cards from a deck of 52 cards is:
C = C = 52!52!
5!(52 – 5)!5!(52 – 5)!
= 2,598,960= 2,598,960
52 5
ExamplesExamples
Your English teacher has asked you Your English teacher has asked you to select 3 novels from a list of 10 to to select 3 novels from a list of 10 to read as an independent project. In read as an independent project. In how many ways can you choose how many ways can you choose which books to read?which books to read?
AnswerAnswer
C C 10!10!
3!(10 – 3)!3!(10 – 3)!
= 120= 120
10 3
Comb. problemComb. problem
Your friend is having a party and has Your friend is having a party and has 15 games to choose from. There is 15 games to choose from. There is enough time to play 4 games. In enough time to play 4 games. In how many ways can you choose how many ways can you choose which games to play?which games to play?
13651365
Combination ProblemCombination Problem
How many ways can two slices of How many ways can two slices of pizza be chosen from a plate pizza be chosen from a plate containing one slice each of containing one slice each of pepperoni, sausage, mushroom, and pepperoni, sausage, mushroom, and cheese pizza.cheese pizza.
AnswerAnswer
C C 4!4!
2!(4 – 2)!2!(4 – 2)!
= 6= 6
4 2
Combination ProblemCombination Problem
How many ways can three colors be How many ways can three colors be chosen from blue, red, green, and chosen from blue, red, green, and yellow?yellow?
AnswerAnswer
C = 4!C = 4!
3!(4 – 3)!3!(4 – 3)!4 3
There are 4 ways to choose three colors from a list of 4 colors.
Comb. ProblemComb. Problem
Find the number of ways two Find the number of ways two
co-chairpeople can be selected for a co-chairpeople can be selected for a committee of 9 people.committee of 9 people.
Answer: 36Answer: 36
Angles in a polygon:Angles in a polygon: http://www.learner.org/channel/cours
es/learningmath/measurement/session4/part_b/sum.html