In an examination, there are 5 multiple choice questions with 3 choices, out of which exac…

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 marks if the question is not attempted. Then the number of ways a student appearing in the examination gets 5 marks is \_\_\_

Answer: 40

Accepted Answer

Let $x$ = correct answers, $y$ = incorrect answers, with $3x-2y=5$ and $x+y\le5$.

The only solution in non-negative integers with $x+y\le5$ is $(x,y)=(3,2)$.

For each correct answer, there's only 1 way (the correct choice); for each incorrect answer, there are 2 wrong choices to pick from.

$$\binom{5}{3}\times1^3\times2^2 = 10\times4 = 40$$

Solutions · 1