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 he can select a book is 63, the value of n is

• A. 2
• B. 3
• C. 4
• D. None

Answer 1

Answer: (B)

3

Answer 2

Since the student is allowed to select at the most n books out of (2n + 1) books, therefore he can select, one book, two books or at the most n books. Hence the number of selecting at least one book is

²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + … + ^{2n + 1}C_n = S = 63

Again we know that

²ⁿ⁺¹C₀ + ²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + … + ²ⁿ⁺¹C_2n + ²ⁿ⁺¹C_2n+1 = 2²ⁿ⁺¹

Now ²ⁿ⁺¹C₀ = ²ⁿ⁺¹C_2n+1 = 1, ²ⁿ⁺¹C₁ = ²ⁿ⁺¹C_2n etc.

Hence we have 1 + 1 + 2S = 2²ⁿ⁺¹

or 2 + 2 . 63 = 2²ⁿ⁺¹

or 128 = 2²ⁿ⁺¹ or 2⁷ = 2²ⁿ⁺¹

\ 2n + 1 = 7

or 2n = 6 \ n = 3.