Question

In an examination, a question paper consists of 12 questions divided into two parts i.e., part I and part II containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Answer 1

Here number of questions in part I are 5 and number of questions in part II are 7. We have to select 8 questions at least 3 questions from each section. So we have required selections are 3 from part I and 5 from part II or 4 from part I and 4 from part II or 5 from part I and 3 from part II.

$\therefore$ Number of ways of selection

= ${}^5{C}_3{\times }^7{C}_5{+}^5{C}_4{\times }^7{C}_4{+}^5{C}_5{\times }^7{C}_3$

= $\frac{5!}{3!2!}\times \frac{7!}{5!2!}+\frac{5!}{4!1!}\times \frac{7!}{4!3!}+\frac{5!}{5!0!}\times \frac{7!}{5!0!}$

= $\frac{5\times 4\times 3!}{3!\times 2\times 1}\times \frac{7\times 6\times 5!}{5!\times 2\times 1}+\frac{5\times 4!}{4!\times 1}\times \frac{7\times 6\times 5\times 4!}{4!\times 3\times 2\times 1}+1\times \frac{7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!}$

= $10\times 12+5\times 35+1\times 35$

= $210+175+35=420.$