Combinations Lesson 4.7

Permutations Revisited Remember that a permutation is an arrangement or listing of objects. P(6, 3) means “how many ways can you line up 3 out of 6 objects” P(n,r)= n! (n-r)! 6! (3)! = 120 P(6,3)= 6! (6-3)! =

Definitions Combination – an unordered selection of elements from a given set. Direct reasoning – all suitable outcomes are totaled to arrive at a final answer. Indirect reasoning – the undesired outcomes are subtracted from the total to arrive at a final answer.

Finding The Number of Combinations Since order is not important, we divide by the number of ways of re-arranging the numbers in the combination C(n,r)= n! (n-r)! r! P(n,r) = n choose r is equal to n permute r, divided by the number of ways of re-arranging the r items

Combination Notation n C(n,r)= nCr = r There are 3 different notations for “n choose r” C(n,r)= nCr = n r

Perms versus Combs r!×C(n,r)= P(n,r) We use permutations for lists of items – when order is important We use combinations for sets of items – where order is not important Several permutations may all represent the same combination There are fewer combinations than permutations r!×C(n,r)= P(n,r)

Ex. #1:Calculate C(5,3)

Ex. #2: How many different sampler dishes with 3 different flavours could you get at an ice-cream shop with 31 different flavours?

Ex. #3: a) You have 30 people, and you must choose 5 for a leadership opportunity. b) Jill must be included in the leadership opportunity.

Ex. #4: How many ways can 6 people be selected from a group that consists of four adults and 8 children if the group must contain exactly two adults?

Direct Versus Indirect Reasoning Ex. #5: How many ways can 6 people be selected from a group that consists of four adults and 8 children if the group must contain at least two adults? Solution 1: Direct Reasoning

Direct Versus Indirect Reasoning Ex. #5: How many ways can 6 people be selected from a group that consists of four adults and 8 children if the group must contain at least two adults? Solution 2: Indirect Reasoning

Probabilities Using Combinations Ex. #6: Five cards are dealt at random from a deck of 52 playing cards. Determine the probability you will have following: a) 5 black cards in your hand. b) 4 of a kind

Probabilities Using Combinations Ex. #6: Five cards are dealt at random from a deck of 52 playing cards. Determine the probability you will have following: b) 4 of a kind

Practice Questions Worksheet 4.7 online: Page 262 #1-5, 7, 9, 10-16