A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is __________.

Solution

Maths sir · 3 steps

Pick which four are right, then count how many ways the other two can go wrong.

1. Choose the correct four. Any 4 of the 6 questions can be the correctly answered ones. That gives C(6, 4) = 15 choices.
2. Fill in the answers. Each of those 4 questions has exactly 1 correct option, so there is 1 way for each. Each of the remaining 2 questions must be answered wrong, and it has 3 wrong options to choose from.

   C(6, 4) × 1⁴ × 3² = 15 × 9

3. So. The number of ways is 15 × 9 = 135.

The common slip: 240. Letting the other two questions take any of the 4 options gives 15 × 16 = 240. That also counts cases with 5 or 6 correct, so those two questions get only 3 options each.

The idea to keep. Exactly r correct means the rest must be wrong. Choose the correct ones with C(n, r), then give each remaining question only its wrong options.

Answer from the NTA final answer key. Solution by PrepWiser.