3.8 Counting Techniques: Combinations

If you are dealt a hand in poker (5 cards), does it matter in which order the cards are dealt to you? A  K  J 10 Q  A  J 10 Q  K  10 Q  K  A  J K  J 10 Q  A  Are these hands the same or different? The same: order doesn’t matter

Combinations Permutation: –Arrangement of r objects (out of n) in which the order matters –E.g.: choosing first, second, third place Combination: –Number of ways of choosing r objects from a set of n when order doesn’t matter –E.g.: choosing a group of 3 people

What’s the pattern? Choosing 3 objects from, A, B, C, D, E ABCABD ABE ACDABCABD ABE ACD BCA BDABEACDABCA BDABEACDA ACB ADBAEBADC etc. There are P(5,3) ways of arranging 3 objects from 5 There are 3! ways of arranging those three objects

However, those 3! ways are “identical” –CAB is just ABC in a different order So we have ways to choose 3 objects from 5: There are 10 ways to choose 3 objects from a set of 5.

Combinations We say “n choose r” We write C(n, r) or Note: it is NOT

Example 1a How many ways can you choose a president and vice-president from a group of 5 people? Does order matter? Yes! We use permutations There are 20 ways to choose a president and vice-president from a group of 5 people.

Example 1b How many ways can you choose a committee of 2 from a group of 5 people? Does order matter? No! We use combinations There are 10 ways to choose a committee of 2 from a group of 5 people.

Example 2 You can also use the n C r button on your calculator. How many ways can we choose 30 objects from a group of 100? Error on the calculator! Type: 100 nCrnCr 30 =

Example 3 How many ways can you choose a hand of 5 cards from a regular deck… a) with no restrictions? b) so that exactly one card is an ace? c) so that all 5 are hearts? d) so that you have 4 of a kind?

Example 3 sol’ns a) no restrictions: b) exactly one card is an ace: First we choose the ace: Then we choose the rest of the cards: So the number of ways is

Example 3 sol’ns c) all 5 are hearts: How many hearts? d) 4 of a kind: Choose the value first: Then choose the suits: Choose 5 of those cards: Then choose the last card: So number of ways to get 4 of a kind: 13

0! What is 0! ? We define 0! = 1 How many ways can you select 0 people from a group of 4? 1 Check:

Types of Reasoning When working out problems, two types of reasoning can be used Direct reasoning –All suitable outcomes are totaled Indirect reasoning –All undesired outcomes are subtracted from total Why would we do this? Sometimes the calculations are easier!

Example 4 In how many ways can you pick 5 people from a group of 6 adults and 8 children if the group must contain at least 2 adults? This means we have a group with –exactly 2 adults –or a group with exactly 3 adults –or a group with exactly 4 adults –or a group with exactly 5 adults !

Direct: # ways to have at least 2 adults = # ways to have exactly 2 adults + # ways to have exactly 3 adults + etc. Example 4 solution Indirect: # ways to have at least 2 adults = total # of groups of 5 – # groups with no adults – # of groups with 1 adult

Example 5 What is the probability of a specific set of numbers winning Lotto 6/49? We choose 6 numbers from a set of 49 For those of you planning your future, this is very, very small.

Poker What is the probability of being dealt the following hands: –1 pair (2 of a kind, 3 different)? –2 different pairs (2 of a kind, 2 of a different kind, 1 different)? –3 of a kind (3 the same and 2 different)? –Straight? (a run of consecutive values, at least one card is a different suit (A can be high or low)) –Flush? (all five cards are of the same suit, but not all consecutive)

Poker What is the probability of being dealt the following hands: –Full house? (1 pair and 3 of a kind) –4 of a kind? –Straight flush? (a run of cards of the same suit, but not 10, J, Q, K, A) –Royal flush? (10, J, Q, K, A of the same suit) –None of the above?