Question

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways in which the candidate can choose these questions is

Answer 1

2250

Answer 2

Let $n_1, n_2, n_3$ be the number of questions chosen from the 3 sections. The conditions are: $n_1 + n_2 + n_3 = 5$ and $n_i \ge 1$ for $i=1, 2, 3$. The possible integer distributions $(n_1, n_2, n_3)$ are permutations of (3, 1, 1) and (2, 2, 1).

For permutations of (3, 1, 1): $3 \times \binom{5}{3}\binom{5}{1}\binom{5}{1} = 3 \times 10 \times 5 \times 5 = 750$. For permutations of (2, 2, 1): $3 \times \binom{5}{2}\binom{5}{2}\binom{5}{1} = 3 \times 10 \times 10 \times 5 = 1500$. Total ways = $750 + 1500 = 2250$.