In an examination, a candidate is required to answer 5 questions in all, from…
2023
In an examination, a candidate is required to answer 5 questions in all, from 2 sections having 5 questions each. What is the total number of ways in which a candidate can select the questions, provided that at least two questions are to be attempted from each section?
- A.
200
- B.
20
- C.
100
- D.
10
Show answer & explanation
Correct answer: A
Concept: nCr = n!/(r!(n-r)!) gives the number of ways to choose r items from a group of n. When a fixed number of items must be picked from two independent groups subject to a joint "at least" condition on each group, list every distribution (a from the first group, b from the second) that satisfies the condition, compute each distribution's count as the product of the two groups' combination counts, and sum across all valid distributions.
The exam has two sections of 5 questions each, and 5 questions must be answered in total, with at least 2 coming from each section.
List the splits (a, b) with a + b = 5 where both a >= 2 and b >= 2: only (2, 3) and (3, 2) qualify — (0,5), (1,4), (4,1) and (5,0) are excluded because one side falls below 2.
For the split (2, 3): C(5,2) x C(5,3) = 10 x 10 = 100 ways.
For the split (3, 2): C(5,3) x C(5,2) = 10 x 10 = 100 ways.
Total ways = 100 + 100 = 200.
Cross-check: with no section restriction, choosing any 5 of the 10 questions gives C(10,5) = 252 ways. Removing the splits that break the "at least two per section" rule — (0,5): C(5,0) x C(5,5) = 1, (1,4): C(5,1) x C(5,4) = 25, (4,1): 25, (5,0): 1, totalling 52 — leaves 252 - 52 = 200, confirming the result.