Question

The greatest coefficient in the expansion of $\left(x+\frac{1}{x}\right)^{2 n}$ is

• A. $\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1) \cdot 2^{n}}{n !}$
• B. $\frac{2 n !}{(n !)^{2}}$
• C. $\frac{n !}{\left[\left(\frac{n}{2}\right) !\right]^{2}}$
• D. none of these

Answer 1

Answer: (A), (B)

Since the middle term has greatest coefficient,
$\therefore $ greatest coefficient $=$ coefficient of the middle term
$={ }^{2 n} C_{n}=\frac{(2 n) !}{n ! n !} $
$=\frac{2 n(2 n-1)(2 n-2)(2 n-3) \ldots 4 \cdot 3 \cdot 2 \cdot 1}{n ! n !}$
$=\frac{[(2 n-1)(2 n-3) \ldots 3 \cdot 1][2 n(2 n-2)(2 n-4) \ldots 4 \cdot 2]}{n ! n !} $
$=\frac{[1 \cdot 3 \cdot 5 \ldots(2 n-1)] 2^{n}[n(n-1)(n-2) \ldots 2 \cdot 1]}{n ! n !} $
$=\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1) 2^{n} n !}{n ! n !}=\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1) 2^{n}}{n !}$