Question

A student is allowed to select utmost n books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select one book is 63, then the value of n is

• A. 2
• B. 3
• C. 4
• D. None of these

Answer 1

Answer: (B)

3

Answer 2

Since the student is allowed to select utmost n books out of $(2n + 1)$ books. Therefore in order to select one book he has the choice to select one, two, three,......., n books.

Thus, if T is the total number of ways of selecting one book then $T =^{2n + 1} ⥂ C_{1} +^{2n + 1} ⥂ C_{2} + ..... +^{2n + 1} ⥂ C_{n} = 63$.

Again the sum of binomial coefficients

$2n + 1C_{0} +^{2n + 1}C_{1} +^{2n + 1} ⥂ C_{2} + ...... +^{2n + 1} ⥂ C_{n} +^{2n + 1} ⥂ C_{n + 1}$+ $2n + 1C_{n + 2} + ..... +^{2n + 1} ⥂ C_{2n + 1} = (1 + 1)^{2n + 1} = 2^{2n + 1}$or, $2n + 1 ⥂ C_{0} + 2(^{2n - 1} ⥂ C_{1} +^{2n + 1} ⥂ C_{2} + ..... +^{2n + 1} ⥂ C_{n}) +^{2n + 1}C_{2n + 1} = 2^{2n + 1}$

⇒ $1 + 2(T) + 1 = 2^{2n + 1}$ ⇒ $1 + T = \frac{2^{2n + 1}}{2} = 2^{2n}$ ⇒ $1 + 63 = 2^{2n}$ ⇒ $2^{6} = 2^{2n}$ ⇒ $n = 3$.