Question

A student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which be can select a book is $255$, then the value of $n$ equals to

• A. $6$
• B. $5$
• C. $4$
• D. $3$

Answer 1

The Correct Option is C

By the given condition $^{2n+1}C_{1}+\,^{2n+1}C_{2}+ ..........+\,^{2n+1}C_{4}=255$ Now $^{2n+1}C_{1}=\,^{2n+1}C_{2n} \, \left[\because \,^{n}C_{r}=\,^{n}C_{n-r}\right]$ $^{2n+1}C_{2}=\,^{2n+1}C_{2n-1}$ $^{2n+1}C_{n} =\,^{2n+1}C_{n+1}$ Adding these, we get $^{2n+1}C_{0}+\,^{2n+1}C_{1}+\,^{2n+1}C_{2}+...........+\,^{2n+1}C_{n}$ $=^{2n+1}C_{n+1}+\,^{2n+1}C_{2n}+\,^{2n+1}C_{2n+1}$ $\Rightarrow 2\left[^{2n+1}C_{0}+\,^{2n+1}C_{1}+..........+\,^{2n+1}C_{n}\right]$ $=2^{2n+1}$