Question

A question paper has two sections A and B. Section A has 8 questions and Section B has 6 questions. A student has to answer 10 questions, choosing at least 4 from section A and at least 3 from section B. The number of ways the student can answer the paper is:

• A. \(800\)
• B. \(820\)
• C. \(840\)
• D. \(986\)

Hint:

Always list valid (A,B) pairs satisfying constraints, then use combinations \( \binom{n}{r} \) to calculate total selections.

Answer 1

The Correct Option is D

Step 1: Identify valid combinations of questions.
Let \( a \) be the number of questions from Section A, and \( b \) from Section B. Then: \[ a + b = 10, \quad a \geq 4, \quad b \geq 3 \] Valid combinations: \( (4,6), (5,5), (6,4), (7,3), (8,2) \)
Step 2: Calculate combinations for each case. \[ \begin{aligned} (4,6):\quad & \binom{8}{4} \cdot \binom{6}{6} = 70 \cdot 1 = 70
(5,5):\quad & \binom{8}{5} \cdot \binom{6}{5} = 56 \cdot 6 = 336
(6,4):\quad & \binom{8}{6} \cdot \binom{6}{4} = 28 \cdot 15 = 420
(7,3):\quad & \binom{8}{7} \cdot \binom{6}{3} = 8 \cdot 20 = 160
(8,2):\quad & \binom{8}{8} \cdot \binom{6}{2} = 1 \cdot 15 = 15 \end{aligned} \]
Step 3: Add all possible ways. \[ 70 + 336 + 420 + 160 + 15 = \boxed{986} \]