Multichoose

Wolfram Alpha defines multichoose as follows, “The number of multisets of length \(k\) on \(n\) symbols is sometimes termed \(n\) multichoose \(k\), denoted \(\left(\binom{n}{k}\right)\) by analogy with the binomial coefficient \(\binom{n}{k}\) \(n\) multichoose \(k\) is given by the simple formula \(\left(\binom{n}{k}\right)=\binom{n+k-1}{k}=\left(n-1,k\right)!\), where \((n-1,k)\) is a multinomial coefficient” (Wolfram).

What is so special about multichoosing is that it can be used for counting problems where order does not matter and repetition is allowed.

Another term one might often hear used in place of a multichoose problem is a “stars and bars” problem. In fact, if you’re ever in a pinch you should be able to come up with the multichoose formula just from the stars and bars method.

It works something like this. Say we are given 3 distinct candies