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Ques 1 GATE 2025 SET-1
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?
(b) is the correct answer.
Explanation:
1. Identify the Type of Problem:
• There are n = 4 distinct flavors available.
• We need to choose r = 3 scoops.
• Since we can choose the same flavor multiple times, repetition is allowed.
• Since the order of the scoops is inconsequential, this is a combinations problem with repetition (also known as the "Stars and Bars" or "Multichoose" problem).
2. Apply the Formula:
The number of ways to choose r items from a set of n distinct items with repetition allowed is given by the formula:
n + r - 1Cr
3. Substitute the Values:
• n = 4 (flavors)
• r = 3 (scoops)
Number of ways = 4 + 3 - 1C3 = 6C3
4. Calculate the Combination:
6C3 = (6 × 5 × 4) / (3 × 2 × 1) = 120 / 6 = 20
5. Conclusion: There are exactly 20 different ways to purchase the 3 scoops of ice-cream.