-r-0n r2 - nCr2 is equal to

∑^n_r=0 r^2 . (^nCr)^2 = n^2 ∑_r=1^n(^n 1C_r 1)^2 = n^2×co efficeint of x^n 1 in (1 x)^2n 2 = n^2 · ^2n 2C_n 1 n^2 . (2n 2)!/((n 1)!)^2 = n^2 . (n 1)! . ...

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Byju's Answer

Standard XIII

Mathematics

Sum of Coefficients of All Terms

∑r=0n r2 . nC...

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 option is C $\frac{n^{2} . 2^{n - 1} (r = 1 \prod n - 1 (2 r - 1))}{(n - 1)!}$
$n \sum r = 0 r^{2} . (^{n} C r)^{2} = n^{2} n \sum r = 1 {(^{n - 1} C_{r - 1})}_{r - 1}^{2}$

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

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

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

Suggest Corrections

n∑r=0r2.(nCr)2 is equal to

The correct option is C n2 . 2n−1(r=1∏n−1 (2r−1))(n−1)!n∑r=0 r2.(nCr)2=n2 n∑r=1(n−1Cr−1)2 =n2×co-efficeint of xn−1 in (1−x)2n−2 =n2⋅ 2n−2Cn−1 n2.(2n−2)!((n−1)!)2=n2.(n−1)!.2n−1.(n−1∏r=0(2r−1))((n−1)!)2 =n2.2n−1.(n−1∏r=0(2r−1))(n−1)!.

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