We define and enumerate combinations.


Combinations are counted by counting permutations and then dividing appropriately.

(problem 1) In how many ways can a 5 person committee be formed from a group of 25 employees?

Each combination of the objects selected from a set of distinct objects corresponds to permutations of those objects. This is because there are ways to permute the selected objects. Thus the number of permutations, is times greater than the number of combinations, . Since we know the number of permutations is we have Dividing both sides by gives the desired result, namely
(problem 3) Compute each of the following:

(problem 4) An NHL hockey team consists of 23 players. A line-up consists of 6 players. How many line-ups are possible (without regard to position)?
(problem 5) How many subsets of size can be made from a set containing elements ()?
How does this answer differ from the answer to the example above?
(problem 6) A box of crayons contains 64 different colors. In how many ways can Jill select 10 different colors to make her drawing?
(problem 7) There are variants of poker in which a player is dealt 7 cards. How many -card poker hands are possible?
(problem 8) How many of each of the following poker hands are possible? a) Four of a kind:
b) Flush (all 5 cards of the same suit):
c) Full house (three of one kind and two of another kind):
d) Two pair: