3.2 Combinations

Combinations Counting where order doesn’t matter. Combination Formula: nCr = n!/r!(n-r)! Example: We have a deck of cards and we want 5 cards. They can be any cards. There are 52 cards to a deck. 52C5 52!/5!(52-5)! = 2,598,960

K selections are made from n items, without regard to order and repeats are allowed. The number of ways these selections can be made is n+K-1CK

Example Number of CD’s to choose from = 10 K = 3 (the number of CD’s that can be ordered) 10+3-1C3 = 12C3 = 12!/3!(12-3)! = 12!/3!9! = 479001600/(6)(362880) = 479001600/2177280 = 220

When order doesn’t matter, we count the number of subsets or combinations. When order matters, we count the number of sequences or permutations. How many different 7 person committees can be formed, each containing three women from an available set of 20 women, and four men from an available set of 30 men.

Task 1: Choose 3 women from the set of 20 women Task 2: Choose 4 men from the set of 30 men 30C4 30!/4!(30-4)! = 30!/4!26! = 27,405 (1140)(27405) = 31,241,700 different committees