Question

The value of

ⁿC₀ .²ⁿC_r – ⁿC₁ .^{2n – 2}C_r + ⁿC₂ .^{2n – 4}C_r +......... + (–1)ⁿⁿC_n^{2n – 2n}C_n is –

• A. $\left\{ \begin{matrix} {2^{2n - r}}^{n}C_{r - n} & , & ifr > n \\ 0 & , & ifr < n \end{matrix} \right.\$
• B. $\left\{ \begin{matrix} 3^{2n - rn}C_{r - n} & , & ifr > n \\ 0 & , & ifr < n \end{matrix} \right.\$
• C. $\left\{ \begin{matrix} {2^{2n - r}}^{n}C_{r + n} & , & ifr > n \\ 0 & , & ifr < n \end{matrix} \right.\$
• D. None of these

Answer 1

Answer: (A)

$\left\{ \begin{matrix} {2^{2n - r}}^{n}C_{r - n} & , & ifr > n \\ 0 & , & ifr < n \end{matrix} \right.\$

Answer 2

We have

ⁿC₀ . ²ⁿC_r – ⁿC₁ . ^2n–2C_r + ⁿC₂ . ^2n–4C_r + ....+ (–1)^{n n}C_n ^2n–2nC_n

= Coefficient of x^r in [ⁿC₀ (1 + x)²ⁿ – ⁿC₁ (1 + x)^2n–2 + ⁿC₂

(1 + x)^2n–4 – ….…+ (–1)^{n n}C_n (1 + x)^2n–2n]

= Coefficient of x^r in [(1 + x)² – 1]ⁿ

= Coefficient of x^r in (2x + x²)ⁿ

= Coefficient of x^r–n in (2 + x)ⁿ

= $\left\{ \begin{matrix} 2^{2n - rn}C_{r - n} & , & ifr > n \\ 0 & , & ifr < n \end{matrix} \right.\$

Hence (1) is correct answer.