1. Combinations
Lesson 4.7

2. 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)! =

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.

4. 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

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

6. 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)

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

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

9. 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.

10. 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?

11. 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

12. 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

13. 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

14. 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

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