Question

A person is permitted to select at least one and at most n coins from a collection of (2n + 1) distinct coins. If the total number of ways in which he can select coins is 255, then n equals

• A. 4
• B. 8
• C. 16
• D. 32

Answer 1

Answer: (A)

4

Answer 2

Since the person allowed to select at most n coins out of (2n + 1) coins, therefore in order to select one, two, three, .............,n coins. Thus if T is the total number of ways of selecting one coin, then

T = ²ⁿ⁺¹ C₁ + ²ⁿ⁺¹C₂ + ............+ ²ⁿ⁺¹C_n = 255

Again the sum of binomial coefficients

= ²ⁿ⁺¹C0 + ²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ........+ ²ⁿ⁺¹C_n + ²ⁿ⁺¹C_n+1 + ²ⁿ⁺¹C_n+2 + ..........+ ²ⁿ⁺¹C_2n+1 = (1 + 1)²ⁿ⁺¹ = 2²ⁿ⁺¹

⇒ ²ⁿ⁺¹C₀ + 2(²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + .........+ ²ⁿ⁺¹C_n) + ²ⁿ⁺¹C_2n+1

= 2²ⁿ⁺¹

⇒ 1 + 2(T) + 1 = 2²ⁿ⁺¹ ⇒ 1 + T = $\frac{2^{2n + 1}}{2} = 2^{2n}$

⇒ 1 + 255 = 2²ⁿ ⇒ 2²ⁿ = 2⁸ ⇒ n = 4.