Ice-cream cups with three scoops

Text

An ice-cream shop offers $6$ flavours. In how many ways can a cup with $3$ scoops be composed, if flavours may repeat and the order of the scoops does not matter?

Solution

Since flavours may repeat and order is irrelevant, these are combinations with repetition of $n=6$ flavours taken $k=3$ at a time: $N = \binom{n+k-1}{k} = \binom{6+3-1}{3} = \binom{8}{3} = \frac{8\cdot 7\cdot 6}{6} = \boxed{56}.$