In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, -2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is ____________.

Solution

Maths sir · 4 steps

First find how many right and wrong answers give 5 marks, then count the choices for each wrong one.

1. Set up marks. Let c be the number of correct answers and w the number of wrong ones, with c + w ≤ 5. Unattempted questions add 0, so the marks are 3c - 2w.

   3c - 2w = 5, c + w ≤ 5

2. Only one split. 3c = 5 + 2w must be odd, so c is odd. c = 1 gives w = -1, which is impossible. c = 5 gives w = 5, which needs 10 questions. Only c = 3, w = 2 works, with 0 unattempted.
3. Count the ways. Choose which 3 of the 5 questions are correct: C(5, 3) = 10. A correct question has only 1 option, but each wrong question has 2 wrong options to pick from, giving 2 × 2 = 4.
4. So. The number of ways is 10 × 4 = 40.

The common slip: 10. Counting only which questions are right and which are wrong gives 10. Each wrong answer can be either of 2 wrong options, so multiply by 2² = 4.

The idea to keep. In marking-scheme counting, solve the marks equation for whole numbers first. Then pick which questions get each outcome, and multiply by the options available for each outcome.

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