Question:
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 ____________.
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 ____________.
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Let c be the number of correct and w the number of wrong answers; solve 3c - 2w = 5 with c + w at most 5, then count the arrangements.
Updated On: Nov 4, 2026
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Correct Answer: 40
Solution and Explanation
Let the student answer c questions correctly and w incorrectly. Then \(3c - 2w = 5\) with \(c + w \le 5\) and c, w non-negative integers.
From \(c = \frac{5 + 2w}{3}\): w = 0 gives c = 5/3 (no); w = 1 gives 7/3 (no); w = 2 gives c = 3 (valid, c + w = 5); w = 5 gives c = 5 but c + w = 10 > 5 (invalid).
So exactly 3 correct, 2 wrong and 0 unattempted.
Choose which 3 questions are correct: \(\binom{5}{3} = 10\) ways (each has exactly 1 correct choice).
Each of the remaining 2 questions is wrong in \(3 - 1 = 2\) ways: \(2 \times 2 = 4\).
Total = \(10 \times 4 = 40\).
Answer: 40
From \(c = \frac{5 + 2w}{3}\): w = 0 gives c = 5/3 (no); w = 1 gives 7/3 (no); w = 2 gives c = 3 (valid, c + w = 5); w = 5 gives c = 5 but c + w = 10 > 5 (invalid).
So exactly 3 correct, 2 wrong and 0 unattempted.
Choose which 3 questions are correct: \(\binom{5}{3} = 10\) ways (each has exactly 1 correct choice).
Each of the remaining 2 questions is wrong in \(3 - 1 = 2\) ways: \(2 \times 2 = 4\).
Total = \(10 \times 4 = 40\).
Answer: 40
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