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 ______

There are 5 MCQ questions with 3 choices each. Scoring: +3 correct, -2 wrong, 0 unattempted. We need the number of ways to score exactly 5 marks.

Set up equations.

Let $$c$$ = correct, $$w$$ = wrong, $$s$$ = skipped. Then:

$$c + w + s = 5$$

$$3c - 2w = 5$$

Find valid combinations.

From $$3c - 2w = 5$$: testing values of $$c$$:

$$c = 1$$: $$w = -1$$ (invalid)

$$c = 3$$: $$w = 2$$, $$s = 0$$ ✓

$$c = 5$$: $$w = 5$$ (but $$c + w = 10 > 5$$, invalid)

The only valid case is $$c = 3, w = 2, s = 0$$.

Count the number of ways.

Choose which 3 questions are answered correctly: $$\binom{5}{3} = 10$$ ways.

For the remaining 2 wrong answers, each has 2 incorrect choices: $$2^2 = 4$$ ways.

Total = $$10 \times 4 = 40$$

Answer: 40