The coefficient of 1-x in the expansion of 1-xn1-1-xn isA- n -n-1 -n-1 -B- 2 n -n-1 -n-1 -C- 2 n -2 n-1 -2 n-1 -

The correct option is B (2n)!(n 1)!(n+1)!Coefficient of x^ 1 in (1+x)^n(1+1x)^n = nbsp; Coefficient of x^ 1 in x^ n(1+x)^2n = nbsp; Coefficient of x^n 1 in ...

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Standard XII

Mathematics

Binomial Coefficients

The coefficie...

Question

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

\frac{n!}{(n - 1)! (n + 1)!}

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\frac{(2 n)!}{(n - 1)! (n + 1)!}

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\frac{(2 n)!}{(2 n - 1)! (2 n + 1)!}

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\frac{2 n!}{(n - 2)! (n + 2)!}

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Solution

The correct option is B $\frac{(2 n)!}{(n - 1)! (n + 1)!}$
Coefficient of $x^{- 1}$ in $(1 + x)^{n} {(1 + \frac{1}{x})}^{n}$
$=$ Coefficient of $x^{- 1}$ in $x^{- n} (1 + x)^{2 n}$
$=$ Coefficient of $x^{n - 1}$ in $(1 + x)^{2 n}$
Now as we know that coefficient of $x^{r}$ in expansion of $(1 + x)^{n}$ is $^{n} C_{r}$
Hence required coefficient is
$=^{2 n} C_{n - 1}$
$= \frac{(2 n)!}{(n - 1)! (n + 1)!}$

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Similar questions

Q. The coefficient of

\frac{1}{x}

in the expansion of

{(1 + x)}^{n} {(1 + \frac{1}{x})}^{n}

is
(a)

\frac{n!}{[(n - 1)! (n + 1)!]}

(b)

\frac{(2 n)!}{[(n - 1)! (n + 1)!]}

(c)

\frac{(2 n)!}{(2 n - 1)! (2 n + 1)!}

(d) none of these

The (n+1)th term from the end in the expansion of $(2 x - \frac{1}{x})^{3 n}$ is:

Q. The coefficient of

x^{n}

in the polynomial

(x +^{n} C_{0}) (x + 3 \cdot^{n} C_{1}) \dots (x + (2 n + 1) \cdot^{n} C_{n})

Q. The coefficient of

x^{n}

in the polynomial

(x + {^{2 n + 1} C}_{0}) (x + {^{2 n + 1} C}_{1}) (x + {^{2 n + 1} C}_{2}) + . . . . (x + {^{2 n + 1} C}_{n})

The middle term in the expansion of $(1 + x)^{2 n}$ is

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Binomial Coefficients

Standard XII Mathematics

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

The correct option is B (2n)!(n−1)!(n+1)!Coefficient of x−1 in (1+x)n(1+1x)n = Coefficient of x−1 in x−n(1+x)2n = Coefficient of xn−1 in (1+x)2n Now as we know that coefficient of xr in expansion of (1+x)n is nCr Hence required coefficient is = 2nCn−1 =(2n)!(n−1)!(n+1)!

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