A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 8 from part A and 5 from part B, in how many ways can he choose the questions?

Answer: Option A

There 10 questions in part A out of which 8 question can be chosen as = ¹⁰C₈
Similarly, 5 questions can be chosen from 10 questions of Part B as = ¹⁰C₅
Hence, total number of ways,
$$\begin{array}{rl}& {=}^{10}{\text{C}}_8{\times }^{10}{\text{C}}_5\\ & =\frac{10!}{2!\times 8!}\times \frac{10!}{5!\times 5}\\ & =\left\{10\times \frac{9}{2}\right\}\times \left\{\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2\times 1}\right\}\\ & =11340\end{array}$$