Multiple choice

An examination consists of $10$ multiple choice questions, where each question has $4$ options, only one of which is correct. In every question, a candidate earns $3$ marks for choosing the correct option, and $-1$ for choosing a wrong option. Assume a candidate answers all questions by choosing exactly one option for each. Then the number of distinct combinations of answers which can earn the candidate a score from the set $\left {15, 16, 17, 18, 19, 20\right }$ is

  1. $27$
  2. $120$
  3. $3240$
  4. $6480$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let c be correct answers, w be wrong answers. c + w = 10. Score = 3c - w = 3c - (10-c) = 4c - 10. We want 4c - 10 in {15, 16, 17, 18, 19, 20}. 4c in {25, 26, 27, 28, 29, 30}. Since c must be an integer, 4c = 28, so c = 7. Number of ways to get 7 correct = 10C7 * 3^3 = 120 * 27 = 3240.

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