Question

A student is allowed to select at most n books from a collection of $p(x)=a{x}^2+bx+c$ books. If the total number of ways in which he can select one book is 63, then the value of n is equal to :

• A. 2
• B. 3
• C. 4
• D. 1

Answer 1

Answer: (B)

Since the student is allowed to select at most n books out of $(2n+1)$ books, $\therefore$ In order to select one book he has to select one book he has the choice to select one, two, three ... n books, thus if T is the total number of ways selecting one book then $T{=}^{2n+1}{C}_1{+}^{2n+1}{C}_2+.....{+}^{2n+1}{C}_n=63$ ?(i) again the sum of binomial coefficients ${}^{2n+1}{C}_0{+}^{2n+1}{C}_2+.....{+}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}+...$ $={(1+1)}^{2n+1}=2^{2n+1}$ Or ${}^{2b+1}{C}_0+2\left({}^{2n+1}{C}_1{+}^{2n+1}{C}_2+....{+}^{2n+1}{C}_n\right)$ ${+}^{2n+1}{C}_{2n+1}=2^{2n+1}$ $\Rightarrow$ $1+2(T0+1=2^{2n+1}$ $\Rightarrow$ $1+T=\frac{2^{2n+1}}{2}=2^{2n}$ $\Rightarrow$ $1+63=2^{2n}$ $\Rightarrow$ $2^6=2^{2n}$ $\Rightarrow$ $n=3$