Expert: David Hemmer - 8/3/2007

Question
Can you tell me how many unique combinations there are if you have 8 possible items but only 5 items in any combination. for example you have items a,b,c,d,e,f,g,h. You could have a combination like aaabc or abcde or ddcaa but aabca would be a duplicate of the first one and therefore not unique.

Answer
The kind of thing you want to count is called a multiset. You are looking for the number of 5 element multisets from a set of cardinality eight. In general the number of k element multisets from a set of cardinality n is the binomial coefficient

n+k-1 C k.

In your example this is 12 C 5 which is 792.

To read about this more see:

http://en.wikipedia.org/wiki/Multiset