Combinations

The formula for the number of possible combinations of r objects from a set of n objects:



Example: How many different committees of 4 students can be chosen from a group of 15?

There are 1365 different committees.

	If the order doesn't matter, it is a Combination.
	If the order does matter it is a Permutation.

1.) Combinations with repetition

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla. You can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections would be

• {c, c, c} (3 scoops of chocolate)
• {b, l, v} (one each of banana, lemon and vanilla)
• {b, v, v} (one of banana, two of vanilla)

(And just to be clear: There are n=5 things to choose from, and you choose r=3 of them.
Order does not matter, and you can repeat!)

where n is the number of things to choose from, and you choose r of them
(Repetition allowed, order doesn't matter)

(5+3-1)!	=	7!	=	5040	= 35
3!(5-1)!		3!×4!		6×24

2.) Combinations without repetition

where n is the number of things to choose from, and you choose r of them
(No repetition, order doesn't matter)

By: Roxanne Ching