A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
Answer
(B)
Explanation
Since the order does not matter, a purchase is fixed by how many scoops of each flavour are taken, that is, by whole numbers \(a, b, c, d \ge 0\) with \(a + b + c + d = 3\). By the stars and bars formula, the count is \(\binom{3 + 4 - 1}{3} = \binom{6}{3} = 20\). Check by cases: all three of one flavour gives 4 ways, two of one flavour and one of another gives \(4 \times 3 = 12\) ways, and three different flavours gives \(\binom{4}{3} = 4\) ways, a total of 20.
Answer: B