15.3 Permutations and Combinations

OBJ: To solve problems involving permutations and combinations

Permutation (Order Important) Arrangement Picture

• EX: A club with 12 members wants to choose a president, vice-president, and treasurer. The order of choices is important: The order A, B, and C for president, vice-president, and treasurer, respectively, is different from the order B, A, and C for the 3 positions. The number of ways of filling the three offices is:

• 12 • 11 • 10 = 1320

• Each slate of officers is a permutation of 3 people from a set of 12, denoted as 12 P 3.

Combination (Order not impt.) Committee

• If the club only wanted to select a committee of 3 out of its 12 members, order is not important. Each of the following selections: ABC, ACB, BAC, BCA, CAB, CBA are the same. To determine the number of different committees of three that can be chosen you would need to divide the previous total by 6

• 12 • 11 • 10 1320 3 • 2 • 1 = 6 = 220

• Each committee is a combination of 3 people from a set of 12, denoted as 12 C 3.

Permutation(order important)

• Example• 12 P 3 = 12 • 11 • 10

• 12 • 11 • 10 • 9! (12 – 3) !

• = 1320

Formulan P r = n (n-1)(n-2) ... (n-r+1)(n-r)! (n – r)!• = n!__

(n - r )!

Combination(order not important)

Example12 C 3 = 12 • 11 • 10

3 • 2 • 1

= 12 • 11 • 10 (12–3)! (12 – 3)! 3!

= 220

Formula

n C r = n (n-1)(n-2) ... (n-r+1)(n-r)! (n-r)!1 • 2 • 3 . . . r

= n!__ ( n - r )! r!

EX: A company advertises two job openings, one for a copywriter and one for an artist. If 10 people who are qualified for either position apply, in how many ways can the openings be filled?Fundamental Counting Principle

10 • 9 =

90

Permutation Formula10 P 2 = 10!(10 – 2)!10 • 9 • 8! 8!10 • 9 =90

EX: A company advertises two job openings for computer programmers, both with the same salary and job description. In how many ways can the openings be filled if 10 people apply.

Fundamental Counting Principle

10 • 9 2 • 1 =90 2 =45

Combination Formula

10 C 2

10!(10 – 2)! 2!10 • 9 • 8! 8! 2!10 • 9 2 • 1 =90 2 =45

EX: For a raffle, 845 tickets are sold.

a) In how many wayscan four $50 giftcertificates be awarded

845 C 4

845!(845-4)! 4!845 • 844 • 843 • 842

4 • 3 • 2 • 1 2.1 • 1010

21,000,000,000

b) In how many ways can a $100, a $50, a $20m and a $10 gift certificate be awarded?

845 P 4

845!(845-4)!845 • 844 • 843 • 842 5.06 • 1011

506,000,000,000

EX: At the Red Lion Diner, an omelet can be ordered plain or with any or all of the following fillings: cheese, onions, and peppers. How many different kinds of omelets are possible?a) 3 C 3

3! __(3 – 3)! 3!1

b) 3 C 2 3! __(3 – 2)! 2!3

c) 3 C 1

3!_ _(3 – 1)! 1!3

d) 3 C 0 3!__(3 – 0)! 0!1

3 C 3 + 3 C 2 + 3 C 1 + 3 C 0 1 + 3 + 3 + 1 =8