Today in Pre-Calculus• Review Chapter 9 – need a

calculator• Homework• Go over Chapter 8 worksheet

questions

Combinatorics• An arrangement of objects in a specific order or selecting

all of the objects.• An arrangement of objects in which order does not

matter.• Difference between permutations and combinations:

– Combinations: grouping of objects

– Permutation: putting objects in specific places or positions, or selecting all of the objects.

ExampleS1)There are ten drivers in a race. How many

outcomes of first, second, and third place are possible?

2)In a study hall of 20 students, the teacher can send only 6 to the library. How many ways can the teacher send 6 students?

Conditional Probability & Tree Diagrams

Two identical cookie jars are on a counter. Jar A contains 2 chocolate chip and 2 peanut butter cookies, while jar B contains 1 chocolate chip cookie. Selecting a cookie at random, what is the probability that it is a chocolate chip cookie?

Conditional Probability Notation: P(A|B) probability of A given B

P(chocolate chip|jar A)=

P(chocolate chip|jar B)=

P(A|B)=

P(jar A|chocolate chip) =

( )

( )

P A and B

P B

Binomial DistributionLet p be the probability of event A and q be the probability of event A not occurring given n trials. Then the probability A occurs r times is

nCn-rprqn-r

Ex: We roll a fair die four times. What is the probability that we roll:a)All 3’s

b) no 3’s

c) Exactly two 3’s

Binomial Theorem(a + b)n = nC0an + nC1an-1b + nC2an-2b2 + … +

nCn-2a2bn-2 + nCn-1abn-1 + nCnbn

Example: (2x2 – 3y)4 =

Find the x6y5 term in the expansion of (x + 3y)11

Sequences• Arithmetic Sequence: a sequence in which there is a

common difference between every pair of successive terms.

Example: 5,8,11,14

General formula: an = a1 + (n-1)d

• Geometric: a sequence in which there is a common ratio (or quotient) between every pair of successive terms.

Example:

General formula: an = a1r(n–1)

1 1 1 1, , , ,....

2 4 8 16

Explicitly Defined Sequence• A formula is given for any term in the sequence

Example: ak = 2k - 5

Find the 20th term for the sequence

Write the explicit rule for the sequence 55, 49, 43, …

Write the explicit rule for the sequence 5, 10, 20, …

Series• Series: the sum of the terms of a sequence {a1, a2, …,an}

• Written as:

• Read as “the sum of ak from k = 1 to n”.

• k is the index of summation5

1

2k

k

1

n

kk

a

6

3

2k

k

formulas

1

1

1:

1

nn

kk

a rGeometric a

r

1 11

: 2 ( 1)2 2

n

k nk

n narithmetic a a a a n d

100

1

:k

Example k

110

1

1: 28

2

k

k

Example

ExampleWrite the sum of the following series using summation notation:

Example 1: 13 + 17 + 21 + … + 49

Example 2: 1 + 8 + 27 + … + (n+1)3

Example 3: 3, 6, 12, …, 12,288

Infinite SeriesAn infinite series can either:

1) Converge – if, as n increases, the series sum approaches a value (S)

2) Diverge – if as n increases, the series sum does NOT approach a value.

1

1

3? , . 7

5

k

k

Does this series converge If so give the sum

Homework

• Pg 708: 2,4,11,19,21,22

• Pg 715: 13,15,19,21

• Pg 728: 31,33, 45-50

• Pg 787: 55-58, 63,70,77-81odd,83,84,