The value of ⁿC₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ….+ ^n+kC_k is equal to

Q: The value of ⁿC₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ….+ ^n+kC_k is equal to

(A) ^n+k+1C_k

(B) ^n+k+1C_n+1

(C) ^n+kC_n+1

(D) None of these

Sol. We have (1+x)ⁿ + (1+x)ⁿ⁺¹ + ….+ (1+x)^n+k

$\large = (1+x)^n \frac{(1+x)^{k+1}-1}{x} $

$\large = \frac{(1+x)^{n+k+1}-(1+x)^n}{x} $

Equating coefficients of xⁿ

ⁿC_n + ⁿ⁺¹C_n + ….+^n+kC_n = ^n+k+1C_n+1

⇒ ⁿC₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ….+ ^n+kC_k

= ^n+k+1C_k = ^n+k+1C_n+1

Hence (A), (B) are the correct answer.