PERMUTATIONS AND COMBINATIONSHomework: Permutation and Combinations WS

WARM UP

There are 7 green marbles, 4 red marbles, and 2 blue marbles in the bag. Jenny picked a green marble from the bag, without replacement. What is the probability that the next marble picked is also green?

WARM UP- SOLUTION

There are 7 green marbles, 4 red marbles, and 2 blue marbles in the bag. Jenny picked a green marble from the bag, without replacement. What is the probability that the next marble picked is also green?

6/12 or 1/2

COMBINATIONS

Combination- order doesn’t matter. If you are dealt 5 cards from a deck it doesn’t

matter what order you get them, when you pick up your hand you have 1 combination of cards.

A combination is a grouping of the elements from a set in which the order doesn’t matter. In a combination, abc and acb would be

considered the same: The elements are the same in both groups, and the order in which they appear does not matter.

EXAMPLE 1

How many combinations are there of the letters a, b, c and d using all letters?

How many combinations are there using 3 of the letters?

EXAMPLE 1- SOLUTIONS

How many combinations are there of the letters a, b, c and d using all letters? There is 1 combination.

How many combinations are there using 3 of the letters? abc, abd, acd, bcd There are 4 combinations of 4 letters taken 3 at

a time.

EXAMPLE 2

How many combinations are there of the 4 letters a, b, c and d using 2 letters at a time?

EXAMPLE 2- SOLUTION

How many combinations are there of the 4 letters a, b, c and d using 2 letters at a time?

ab ac ad bc bd cd

There are 6 combinations.

EXAMPLE 3

How many combinations are there of the 4 letters a, b, c and d using 1 letter at a time?

EXAMPLE 3- SOLUTION

How many combinations are there of the 4 letters a, b, c and d using 1 letter at a time?

a b c d

There are 4 combinations.

COMBINATIONS- FORMULA

The combination of n things taken r at a time is

5! is read “five factorial”.It means (5)(4)(3)(2)(1) = 120

nCr n!

r!(n r)!

EXAMPLE 4

Find 10C6

There are lots of factors that you can cross out once you expand your factorials.

10!

6!(10 6)!10!

6!4!109876543216543214321

109874321

10371

210

EXAMPLE 5 Find 6C2 ,9C4 and 10C7.

6C2 =

9C4=

10C7=

EXAMPLE 5- SOLUTIONS Find 6C2 ,9C4 and 10C7.

6C2 =

9C4=

10C7=

6!

2!(6 2)!6!

2!4!654321214321

15

9!

4!(9 4)!9!

4!5!987654321432154321

126

10!

7!(10 7)!10!

7!3!109876543217654321321

120

EXAMPLE 6

There are 6 questions on Elizabeth’s essay test. She only needs to answer 2 of them, she can choose any 2 that she wants. How many different combinations of 2 test questions can Elizabeth answer?

EXAMPLE 6-SOLUTION

There are 6 questions on Elizabeth’s essay test. She only needs to answer 2 of them, she can choose any 2 that she wants. How many different combinations of 2 test questions can Elizabeth answer?

6C2 =

6!

2!(6 2)!654321214321

15

PERMUTATIONS

A permutation is an arrangement of objects in an specific order.

Order matters. $125 is very different that $512

EXAMPLE 7

How many permutations are there using the letters ABC?

EXAMPLE 7- SOLUTIONS

How many permutations are there using the letters ABC?

ABC, ACB, BCA, CBA, BCA, BAC = 6

These are dependent events, and using the fundamental counting principle we get

3 x 2 x 1 or 3!

PERMUTATIONS- FORMULA

The permutations of n things taken r at a time is

nPr n!

(n r)!

EXAMPLE 8 Find 6P2 ,9P4 and 8P5.

6P2 =

9P4=

8P5=

EXAMPLE 8- SOLUTIONS Find 6P2 ,9P4 and 8P5.

6P2 =

9P4=

8P5=

6!

(6 2)!6!

4!6543214321

30

9!

(9 4)!9!

5!987654321

543213,024

8!

(8 5)!8!

3!87654321

321

EXAMPLE 9

Determine if each is a permutation or a combination. Assuming that any arrangement of letters forms

a 'word', how many 'words' of any length can be formed from the letters of the word MATH?

Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter.

How many ways are there to select a subcommittee of 7 members from among a committee of 17?

EXAMPLES-SOLUTIONS

Determine if each is a permutation or a combination. Assuming that any arrangement of letters forms

a 'word', how many 'words' of any length can be formed from the letters of the word MATH?

Permutation Find the number of ways to take 20 objects and

arrange them in groups of 5 at a time where order does not matter.

Combination How many ways are there to select a

subcommittee of 7 members from among a committee of 17?

Combination