Question

The coefficient of the middle term in the expansion of (1 + x) ²ⁿ is

(a) \(\frac{2^n(2n-1)(2n-3)...3.1}{n(n-1)(n-2)...3.2.1}\)

(b) n (n + 1) (n + 2) ..... 2n

(c) 2ⁿ (2n – 1) (2n – 3) ..... 3. 1 

(d) \(\frac{2^n (n) (n+1) (n+2) ... (2n)}{n (n-1) (n-2) ... 3.2.1}\)

Answer 1

Answer : (a) _{\(\frac{2^n [(2n-1) (2n-3) ... 3.1]}{[n (n-1) (n-2) ... 3.2.1]}\)}

The middle term in the expansion of (1 + x)²ⁿ is \((\frac{2n}{2} + 1)\)th term, i.e., (n+1)th term.

∴ Coefficient of (n + 1)th term in the expansion of (1 + x) ²ⁿ = ²ⁿC_n

_{= \(\frac{2^n [(2n-1) (2n-3) ... 3.1]}{[n (n-1) (n-2) ... 3.2.1]}\)}