Question

A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?

Answer 1

The various possibilities for answering the 10 questions are given below:
(i) 4 from part A and 6 from part B.
(ii) 5 from part A and 5 from part B.
(iii) 6 from part A and 4 from part B.
∴ Required number of ways = ${}^6{C}_4{\times }^7{C}_6+{}^6{C}_5{\times }^7{C}_5+{}^6{C}_6{\times }^7{C}_4$

$$\begin{gathered} =\frac{6!}{4!2!}\times 7+6\times \frac{7!}{5!2!}+1\times \frac{7!}{4!3!} \\ =105+126+35 \\ =266 \end{gathered}$$