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7.Binomial Theorem
hard
The coefficient of $\frac{1}{x}$ in the expansion of ${(1 + x)^n}{\left( {1 + \frac{1}{x}} \right)^n}$ is
A
$\frac{{n!}}{{(n - 1)!(n + 1)!}}$
B
$\frac{{(2n)\,!}}{{(n - 1)!(n + 1)!}}$
C
$\frac{{n!}}{{(n - 1)!(n + 1)!}}$
D
None of these
Solution
(b) ${(1 + x)^n} = {\,^n}{C_0} + {\,^n}{C_1}x + {\,^n}{C_2}{x^2} + …. + {\,^n}{C_n}{x^n}$
${\left( {1 + \frac{1}{x}} \right)^n} = {\,^n}{C_0} + {\,^n}{C_1}\frac{1}{x} + {\,^n}{C_2}\frac{1}{{{x^2}}} + …. + {\,^n}{C_n}{\left( {\frac{1}{x}} \right)^n}$
Obviously, required coefficient of $\frac{1}{x}$ can be given by
$^n{C_0}{\,^n}{C_1} + {\,^n}{C_1}{\,^n}{C_2} + …. + {\,^n}{C_{n – 1}}^n{C_n}$
$ = \frac{{(2n)!}}{{(n – 1)!(n + 1)!}}$
Standard 11
Mathematics
