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 $n=3$, because Reason: ${}^{2n+1}{C}_0{+}^{2n+1}{C}_1+...{+}^{2n+1}{C}_n=4^n$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is D Both Assertion and Reason are incorrect

Let

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

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

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

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

${\Rightarrow }^{2n+1}{C}_1{+}^{2n+1}{C}_2+...{+}^{2n+1}{C}_n=255$

$\Rightarrow 4^n-1=255\Rightarrow 4^n=256=4^4\Rightarrow n=4.$