Question

Assertion :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 he can select at least one book is $255,$ then $n=3$ Reason: because ${}^{(2n+1)}{C}_0{+}^{(2n+1)}{C}_1{+}^{(2n+1)}{C}_2+...{+}^{(2n+1)}{C}_n=4^n$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Let $S{=}^{(2n+1)}{C}_0{+}^{(2n+1)}{C}_1{+}^{(2n+1)}{C}_2+...{+}^{(2n+1)}{C}_n$ ...(1)

Using ${}^n{C}_r{=}^n{C}_{n-r}$, we can write (1) as

$S{=}^{(2n+1)}{C}_{2n+1}{+}^{(2n+1)}{C}_{2n}{+}^{(2n+1)}{C}_{2n-1}+...{+}^{(2n+1)}{C}_{n+1}$

Adding (1) and (2), we get

$2S{=}^{(2n+1)}{C}_0+...{+}^{(2n+1)}{C}_{2n+1}=2^{2n+1}\Rightarrow S=2^{2n}=4^n$

The number of ways of choosing $r$ books out of $2n+1$ is ${}^{(2n+1)}{C}_r$.

We are given

${}^{(2n+1)}{C}_1{+}^{(2n+1)}{C}_2{+}^{(2n+1)}{C}_3+...{+}^{(2n+1)}{C}_n=255\Rightarrow 4^n-1=255\Rightarrow 4^n=256=4^4\Rightarrow n=4$