Question

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?

• A. 4
• B. 20
• C. 24
• D. 48

Hint:

When calculating combinations with repetition, use the formula: \(\binom{n+r-1}{r}\) where \(n\) is the number of distinct items and \(r\) is the number of selections.

Answer 1

The Correct Option is B

This problem is a typical example of combinations with repetition, also known as multiset combinations. We need to find how many ways we can choose 3 scoops of ice cream from 4 distinct flavors, where the order of selection does not matter, and repetition of flavors is allowed. This can be calculated using the formula for combinations with repetition: \[ \binom{n + r - 1}{r} \] where:
\(n\) is the number of distinct items (in this case, 4 flavors of ice-cream),
\(r\) is the number of selections (in this case, 3 scoops of ice-cream). Substituting the values: \[ \binom{4 + 3 - 1}{3} = \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20. \] Therefore, the correct answer is (B) 20.