MDM4U 4.7 Counting Techniques and Probability Strategies

MDM4U 4.7 Counting Techniques and Probability Strategies- Combinations Date ______

When Order is Not Important

A combination is an unordered selection of elements from a set

There are many times when order is not important  Suppose Mr. Russell has 10 basketball players and must choose a starting lineup of 5 players (without specifying positions)

Order of players is not important  We use the notation C(n,r) or nCr where n is the number of elements in the set and r is the number we are choosing Combinations

 A combination of 5 players from 10 is calculated the following way, giving 252 ways for Mr. Russell to choose his starting lineup n n! C(n,r)      r  (n  r)!r! 10 10! C(10,5)      5  (10  5)!5! 10!   252 5!5!

An Example of a Restriction on a Combination

Suppose that one of Mr. Russell’s players is the superintendent’s daughter, and so must be one of the 5 starting players

Here there are really only 4 choices from 9 players  So the calculation is C(9,4) = 126 Now there are 126 possible combinations for the starting lineup Combinations from Complex Sets  If you can choose of 1 of 3 entrees, 3 of 6 vegetables and 2 of 4 desserts for a meal, how many possible combinations are there?  Combinations of entrees = C(3,1) = 3  Combinations of vegetables = C(6,3) = 20  Combinations of desserts = C(4,2) = 6  Possible combinations =  C(3,1) x C(6,3) x C(4,2) = 3 x 20 x 6 = 360  You have 360 possible dinner combinations, so you had better get eating! MDM4U 4.7 Counting Techniques and Probability Strategies- Combinations Date ______

Calculating the Number of Combinations

Suppose you are playing coed volleyball, with a team of 4 men and 5 women The rules state that you must have at least 3 women on the floor at all times (6 players)

 How many combinations of team lineups are there?

You need to take into account team combinations with 3, 4, or 5 women

Solution 1: Direct Reasoning

 In direct reasoning, you determine the number of possible combinations of suitable outcomes and add them  Find the combinations that have 3, 4 and 5 women and add them

45 45 45             33 24 15  410  65  41  40  30  4  74 Solution 2: Indirect Reasoning

 In indirect reasoning, you determine the total possible combinations of outcomes and subtract unsuitable combinations  Find the total combinations and subtract those with 2 women 9 45        6 42  84 110  84 10  74

Example. For your favourite sport…  How many players on the roster?  How many players in the starting lineup?  How many different groups of players can be put in the starting lineup? (no assigned positions)  How many ways can the coach set the starting lineup? (assigned positions)

Solution: Hockey  12F, 6D, 2G = 20 players  3F, 2D, 1G start  There are  C(20,6) = 38 760 groups of 6 players MDM4U 4.7 Counting Techniques and Probability Strategies- Combinations Date ______

 P(12, 3) x P(6, 2) x P(2, 1) = 1320 x 30 x 2 = 79 200 starting lineups Finding Probabilities Using Combinations

What is the probability of drawing a Royal Flush (10-J-Q-K-A from the same suit) from a deck of cards?  There are C(52,5) ways to draw 5 cards  There are 4 ways to draw a royal flush  P(Royal Flush) = 4 / C(52,5) = 1 / 649 740

You will likely need to play a lot of poker to get one of these hands!

What is the probability of drawing 4 of a kind?  There are 13 different cards that can be used to make up the 4 of a kind, and the last card can be any other card remaining 448 13   4 1  1 P      52 4165    5  Probability and Odds  These two terms have different uses in math  Probability involves comparing the number of favorable outcomes with the total number of possible outcomes  If you have 5 green socks and 8 blue socks in a drawer the probability of drawing a green sock is 5/13  Odds compare the number of favorable outcomes with the number of unfavorable  With 5 green and 8 blue socks, the odds of drawing a green sock is 5 to 8 (or 5:8) Combinatorics Summary n!  In Permutations, order matters P(n,r)   e.g., President n  r!

 In Combinations, order doesn’t matter  e.g., Committee n n! C(n,r)      r  (n  r)!r!