Question

If n is odd, then C₀² – C₁² + C₂² – C₃² +……….(–1)ⁿ C_n² equals–

• A. 0
• B. 1
• C. 
• D. $\frac{n!}{\left( \frac{n}{2} \right)^{2}!}$

Answer 1

Answer: (A)

0

Answer 2

(1 + x)ⁿ = C₀ + C₁x + C₂x²+…….C_nxⁿ ––––(i)

(x–1)ⁿ = C₀xⁿ – C₁x^n–1 + C₂x^n–2 ……..(–1)^{n n}C_n–(ii)

Multiplying (i) & (ii) and equating the coeff of xⁿ, we get,

C₀² – C₁² + C₂² …………(–1)ⁿ C_n² = ⁿC_n/2 (–1)^n/2 (x²)^n/2

Above is possible only when $\frac{n}{2}$ is an integer i.e. n is even and in case n is odd, then term xⁿ will not occur