Question

A student is to answer $10$ out of $13$ questions in an examination such that the he must choose at least $4$ from the first five questions. The number of choices available to him is :

• A. $140$
• B. $196$
• C. $280$
• D. $348$

Answer 1

Answer: (B)

$196$

Answer 2

The number of choices available to him is $= \,^{5}C_{4}\times\,^{8}C_{6}+\,^{5}C_{5} \times\,^{8}C_{5}$ $\left[\because \,^{8}C_{6}=\,^{8}C_{2}, \,^5C_4=\,^{5}C_{1}, \,^{8}C_{5}=\,^{8}C_{3}\right]$ $=5\times\frac{8\times7}{1\times2}+1\times\frac{8\times7\times6}{1\times2\times3}$ $= 140 + 56 = 196$