Question

A student is to answer \(10\) out of \(13\) questions in an examination such that he must choose at least \(4\) from the first \(5\) questions. The number of choices available to him is:

• A. \(196\)
• B. \(140\)
• C. \(168\)
• D. \(176\)

Hint:

For selection problems with “at least” conditions, split the problem into separate cases and add the number of ways from all valid cases.

Answer 1

The Correct Option is A

Step 1: Interpret the selection constraints.
Out of 13 questions, the student must answer exactly 10. From the first 5 questions, at least 4 must be chosen. Two cases arise.

Step 2: Case 1 — exactly 4 from the first 5.
Choose 4 from the first 5 in ⁵C₄ = 5 ways. The remaining 6 questions must come from the other 8, giving ⁸C₆ = 28 ways. Total for this case: 5 × 28 = 140.

Step 3: Case 2 — all 5 from the first 5.
Choose all 5 in ⁵C₅ = 1 way. The remaining 5 questions come from the other 8, giving ⁸C₅ = 56 ways. Total: 1 × 56 = 56.

Step 4: Sum the possibilities.
Total number of choices = 140 + 56 = 196.

Step 5: Final conclusion.
There are 196 ways to select the questions.