GATE CS 2025 Set 1 Q10 — General Aptitude

This is a classic problem of combinations with repetition (also known as the stars and bars problem or finding the number of non-negative integer solutions).

We need to choose $r = 3$ scoops of ice-cream from $n = 4$ distinct flavors. Since we can choose multiple scoops of the same flavor and the order of purchase does not matter, we can model this as:

where $x_i \ge 0$ represents the number of scoops chosen of flavor $i$.

The number of non-negative integer solutions is given by the formula:

Calculating the value:

Alternative Method (Case-by-Case Analysis):

Let the four flavors be $A, B, C,$ and $D$. We can categorize the selections as follows:

1. All 3 scoops of the same flavor (e.g., $AAA$):
   There are $\binom{4}{1} = 4$ ways.
2. 2 scoops of one flavor and 1 of another (e.g., $AAB$):
   Select the flavor for 2 scoops in $\binom{4}{1} = 4$ ways, and the flavor for 1 scoop in $\binom{3}{1} = 3$ ways.
   Total ways = $4 \times 3 = 12$ ways.
3. All 3 scoops of different flavors (e.g., $ABC$):
   Select 3 distinct flavors out of 4 in $\binom{4}{3} = 4$ ways.

Adding the ways from all cases:

Thus, there are 20 ways to make the purchase.