GATE CS 2025 Set 1 Q.10: Quantitative Aptitude

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?

$4$

$20$

$24$

$48$

Official answer and solution

Official answer: B
From: Official GATE 2025 answer key

Solution

Use the combinations with repetition (stars and bars) formula $\binom{n + r - 1}{r}$ for choosing $r = 3$ unordered items from $n = 4$ types.

Selecting $r = 3$ scoops from $n = 4$ distinct flavors where repetition is allowed and order does not matter is the number of non-negative integer solutions to $x_1 + x_2 + x_3 + x_4 = 3$ :

\binom{n + r - 1}{r} = \binom{4 + 3 - 1}{3} = \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20

Final answer

Option (B)

Common trap

Using $4^3 = 64$ (ordered selections) or $4 \times 3 \times 2 = 24$ (ordered without repetition).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 10 of 65
Section: General Aptitude

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