Prove that if n identical dice are rolled there are n5 choos

Prove that if n identical dice are rolled there are

n+5 choose 5

different possible outcomes.

Solution

If we paraphrase the question, we ask another similar question which is :

Out of N dice, how many of them will have different numbers on them?

NOw, I assume by distinct outcome you mean a unique Multiset of (let\'s say)10 dice faces which is to say an outcome like {1,1,2,2,3,3,4,4,5,6} is same as {1,2,3,4,5,6,1,2,3,4}.

Now this multiset will have a maximum of six unique elements {1,2,3,4,5,6} but the count for each of the unique element could vary between 0 to 10 such that the total count of elements in the set is 10.

The problem is equivalent to dividing 10 similar items into 6 different boxes, which would be (n+k-1) C (k-1) for n items and k boxes. so,

N+6-1 C 6-1 = N+5 C 5

See beggars method for more details.