- r -0 n r 2- n C r 2 is equal toA- n 2-2 n -2 C n -1B- n 2-2 n C nC - n 2- 2 n -1-n-1 r -12 r -1- n -1 -D -2 n - r -1 n 2 r -1- n -

The correct options are A n^2 . ^2n 2C_n 1 C n^2 . 2^n 1( ∏_n 1^r=1 (2r 1))/(n 1)!∑^n_r=0 r^2 . (^nCr)^2 = n^2 ∑_r=1^n(^n 1C_r 1)^2 = n^2×co efficeint ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

∑ r =0 n r 2·...

Question

$n \sum r = 0 r^{2} . (^{n} C_{r})^{2}$ is equal to

n^{2} .^{2 n - 2} C_{n - 1}

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n^{2} .^{2 n} C_{n}

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\frac{n^{2} . 2^{n - 1} (r = 1 \prod n - 1 (2 r - 1))}{(n - 1)!}

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\frac{2^{n} (n \prod r = 1 (2 r - 1))}{n!}

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Solution

The correct options are
A $n^{2} .^{2 n - 2} C_{n - 1}$
C $\frac{n^{2} . 2^{n - 1} (r = 1 \prod n - 1 (2 r - 1))}{(n - 1)!}$
$n \sum r = 0 r^{2} . (^{n} C r)^{2} = n^{2} n \sum r = 1 {(^{n - 1} C_{r - 1})}_{r - 1}^{2}$

$= n^{2} \times co-efficeint of x^{n - 1} in (1 - x)^{2 n - 2}$
$= n^{2} \cdot^{2 n - 2} C_{n - 1}$

$\frac{n^{2} . (2 n - 2)!}{((n - 1)!)^{2}} = \frac{n^{2} . (n - 1)! . 2^{n - 1} . (n - 1 \prod r = 0 (2 r - 1))}{((n - 1)!)^{2}}$

$= \frac{n^{2} . 2^{n - 1} . (n - 1 \prod r = 0 (2 r - 1))}{(n - 1)!} .$

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n∑r=0r2.(nCr)2 is equal to

$n \sum r = 0 r^{2} . (^{n} C_{r})^{2}$ is equal to