Candies to four children

Text

In how many ways can $12$ identical candies be distributed to $4$ children (a child may receive none)?

Solution

The candies are indistinguishable: only how many each child gets matters. This is the classic “stars and bars” scheme: distributing $k=12$ identical objects into $n=4$ distinct boxes corresponds to combinations with repetition $N = \binom{n+k-1}{k} = \binom{4+12-1}{12} = \binom{15}{12} = \binom{15}{3} = \frac{15\cdot 14\cdot 13}{6} = \boxed{455}.$