Section 6.4

Permutations and Combinations

Permutations

Combinations

A permutation of a set of objects is an arrangement of these objects in a definite order.

A combination is a selection of r objects from a set of n objects where order is not important .r n

n–Factorial

For any natural number n,! ( 1)( 2) ... 3 2 1

0! 1

n n n n

Ex. 5! = 5(4)(3)(2)(1) = 120

This notation allows us to write expressions associated with permutations and combinations in a compact form.

Ex. 7!

5!

7 6 5!

5!

7 6 42

Permutations of n Distinct Objects

The number of permutations of n distinct objects taken r at a time is given by

!, where

!

nP n r r n

n r

Ex. 6!

6,36 3 !

P

6 5 4 3!

3!

6 5 4 120

Ex. A boy has 4 beads – red, white, blue, and yellow. How different ways can three of the beads be strung together in a row?

4!

4,34 3 !

P

4! 24

24 different ways

This is a permutation since the beads will be in a row (order).

total number selected

Permutations of n Objects, Not all Distinct

then number of permutations of these n objects taken n at a time is given by

1 2

!

! !... !r

n

n n n

Given n objects with n1 (non-distinct) of type 1, n2 (non-distinct) of type 2,…, nr (non-distinct) of type r where n = n1 + n2 + … + nr

Ex. How many distinguishable arrangements are there of the letters of the word initializing?

12!

5!2!There are 12 letters

i appears 5 times

n appears 2 times

12! 12 11 10 9 8 7 6 5!

5!2! 5!2!

12 11 10 9 8 7 6

2!

1995840

Combinations of n Objects

The number of combinations of n distinct objects taken r at a time is given by

!, where

! !

nC n r r n

r n r

Ex. Find C(9, 6).

9!

9,66! 9 6 !

C

9 8 7 6!

6! 3!

9 8 7

3!

= 84

Ex. A boy has 4 beads – red, white, blue, and yellow. How different ways can three of the beads be chosen to trade away?

4!

4,33! 4 3 !

C

4!4

3!

4 different ways

This is a combination since they are chosen without regard to order.

total number selected