A shop has 4 distinct flavors of ice-cream; scoops of any flavor can be repeated and order doesn't matter. In how many ways can 3 scoops be purchased?
Options:
(A) 4
(B) 20
(C) 24
(D) 48
Official Answer Key:
B
Step-by-Step Solution & Explanation
• **Concept:** Selecting $r$ items from $n$ distinct categories where items can be repeated and order does not matter is a combinations with repetition problem (Stars and Bars theorem).
• **Formula:** $\binom{n + r - 1}{r} = \frac{(n + r - 1)!}{r!(n - 1)!}$
• **Given:** Number of flavors $n = 4$, scoops to purchase $r = 3$.
• **Calculation:** $\binom{4 + 3 - 1}{3} = \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$ ways.
• **Conclusion:** There are 20 possible combinations of 3 scoops. **Correct Answer: B**.
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