If x m occurs in the expansion of x -1- x 22 n- then the coefficient of x m isA- 2 n -m -2 n-mB- 2 n - 3 - 3 -2 n-m -D- 2 n -2 n-m-3 -4 n-m-3 -2 n-m-3 -3 n-m-2 -

The correct option is C (2n)!(2n m3)!( 4n+m3)!T_r+1 = ^2nC_rx^2n r(1x^2)^r = ^2nC_rx^2n 3r This contains x^m, If 2n 3r = m ⇒ r = 2n m3 Hence, coefficien ...

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Standard XII

Mathematics

Binomial Coefficients

If x m occurs...

Question

If $x^{m}$ occurs in the expansion of ${(x + \frac{1}{x^{2}})}^{2 n}$ , then the coefficient of $x^{m}$ is

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

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\frac{(2 n)! 3! 3!}{(2 n - m)!}

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\frac{(2 n)!}{(\frac{2 n - m}{3})! (\frac{4 n + m}{3})!}

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\frac{(2 n)!}{(\frac{2 n - m}{3})! (\frac{3 n + m}{2})!}

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Solution

The correct option is C $\frac{(2 n)!}{(\frac{2 n - m}{3})! (\frac{4 n + m}{3})!}$
$T_{r + 1} =^{2 n} C_{r} x^{2 n - r} {(\frac{1}{x^{2}})}^{r} =^{2 n} C_{r} x^{2 n - 3 r}$
This contains $x^{m}$ , If
$2 n - 3 r = m$
$\Rightarrow r = \frac{2 n - m}{3}$
Hence, coefficient of $x^{m} =^{2 n} C_{r}$
$= \frac{2 n!}{(2 n - r)! r!}$

$= \frac{2 n!}{(2 n - \frac{2 n - m}{3})! (\frac{2 n - m}{3})!}$

$= \frac{2 n!}{(\frac{4 n + m}{3})! (\frac{2 n - m}{3})!}$

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Similar questions

Q. If

x^{p}

occurs in the expansion of

{(x^{2} + \frac{1}{x})}^{2 n} .

then prove that its coefficient is

\frac{2 n!}{\frac{(4 n - p)!}{3!} \frac{(2 n + p)!}{3!}} .

Simplify and express the result in its simplest form

\sqrt{m^{2} n^{2}} \times \sqrt[6]{m^{2} n^{2}} \times \sqrt[3]{m^{2} n^{2}}

Q. The coefficient of

x^{r} (0 \leq r \leq (n - 1))

in the expansion of

(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}

Q. Prove that,

(^{2 n} C_{1})^{2} + 2. (^{2 n} C_{1})^{2} + 3. (^{2 n} C_{3})^{2} + . . . . + 2 n . (^{2 n} C_{2 n})^{2} = \frac{(4 n - 1)!}{{(2 n - 1)!}^{2}}

Q. If

x^{p}

occurs in the equation

{(x^{2} + \frac{1}{x})}^{2 n}

., prove that its coefficient is

\frac{| 2 n - - -}{| \frac{1}{3} (4 n - p) - --------- - | \frac{1}{3} (2 n + p) - --------- -}

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Binomial Coefficients

Standard XII Mathematics

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If xm occurs in the expansion of (x+1x2)2n, then the coefficient of xm is

The correct option is C (2n)!(2n−m3)!(4n+m3)!Tr+1= 2nCrx2n−r(1x2)r= 2nCrx2n−3r This contains xm, If 2n−3r=m ⇒r=2n−m3 Hence, coefficient of xm= 2nCr =2n!(2n−r)! r! =2n!(2n−2n−m3)!(2n−m3)! =2n!(4n+m3)!(2n−m3)!

If $x^{m}$ occurs in the expansion of ${(x + \frac{1}{x^{2}})}^{2 n}$ , then the coefficient of $x^{m}$ is