GATE2025_CS1_Q10

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?

✅ Final Answer:

The correct answer is option (B) 20.

🔹 Short Answer:

This is a problem of combinations with repetition. We have to choose 3 scoops (items) from 4 distinct flavors (categories), where repetition is allowed and order doesn't matter. The formula is C(n+r-1, r), where n=4 (flavors) and r=3 (scoops). This gives C(4+3-1, 3) = C(6, 3) = (6 * 5 * 4) / (3 * 2 * 1) = 20.

🔸 Long Answer:

This question is a classic problem in combinatorics, specifically dealing with **combinations with repetition**. Let's break down why and how to solve it.

Step 1: Identify the type of problem

• We are choosing a number of items (3 scoops).
• These items are chosen from a set of categories (4 distinct flavors).
• "The order in which the scoops are purchased is inconsequential" means this is a **combination**, not a permutation.
• "One can purchase any number of scoops of any flavor" means **repetition is allowed**. For example, getting 3 scoops of chocolate is a valid choice.

This confirms we need the formula for combinations with repetition, often solved using the "Stars and Bars" method.

Step 2: Apply the Stars and Bars Method

Imagine we have 3 scoops (stars ★) that we want to categorize into 4 flavors. To separate these 4 categories, we need 3 bars (|). For example, let the flavors be Vanilla, Chocolate, Strawberry, and Mango.

A purchase can be represented as a sequence of stars and bars.

• ★ | ★ | ★ | represents 1 Vanilla, 1 Chocolate, 1 Strawberry, 0 Mango.
• ★★★ | | | represents 3 Vanilla, 0 of others.
• | ★★★ | | represents 0 Vanilla, 3 Chocolate, 0 of others.

In total, we have 3 stars (scoops, `r`) and 3 bars (number of flavors - 1, which is `n-1`). This gives us a total of r + (n-1) positions to fill.
Number of positions = 3 + (4-1) = 6.

The problem then becomes: "In how many ways can we choose 3 positions for the stars out of the 6 available positions?" (The remaining positions will automatically be filled by bars).

Step 3: Use the Combination Formula

The number of ways is given by the combination formula C(k, r), where k is the total number of positions and r is the number of items to choose.

C(n + r - 1, r) = C(4 + 3 - 1, 3) = C(6, 3)

Now, we calculate C(6, 3):

C(6, 3) = 6! / (3! * (6-3)!)
= 6! / (3! * 3!)
= (6 × 5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1) × (3 × 2 × 1))
= (6 × 5 × 4) / (3 × 2 × 1)
= 120 / 6
= 20

There are 20 different ways to purchase 3 scoops of ice cream from 4 distinct flavors.

Bibliography:
- Rosen, K. H. (2018). Discrete Mathematics and Its Applications. McGraw-Hill Education. (Chapter 6: Counting).
- Grimaldi, R. P. (2003). Discrete and Combinatorial Mathematics: An Applied Introduction. Pearson.

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