Find the value of n C 0- n -1- n - n C 1- n -1 n C 2- -1 - n C nA- -2nn-1B -2n-1n-2C -2n-1n-2D -2nn-2

The correct option is B Each term of the sum is of the form (n+1 r)^nC_r ⇒ sum = ∑_r=0^n((n+1) r)^nC_r = ∑_r=0^n(n+1)^nC_r ∑_r=0^nr.^nC_r =(n+1)× 2^n ∑_r=1^nn ...

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Byju's Answer

Standard XI

Mathematics

Binomial Coefficients

Find the valu...

Question

Find the value of $^{n} C_{0} . (n + 1) + n .^{n} C_{1} + (n - 1)^{n} C_{2} . . . . 1.^{n} C_{n}$

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Solution

The correct option is B

Each term of the sum is of the form $(n + 1 - r)^{n} C_{r}$
$\Rightarrow s u m = \sum_{r = 0}^{n} ((n + 1) - r)^{n} C_{r}$
$= \sum_{r = 0}^{n} (n + 1)^{n} C_{r} - \sum_{r = 0}^{n} r .^{n} C_{r}$
$= (n + 1) \times 2^{n} - \sum_{r = 1}^{n} n \times^{n - 1} C_{r - 1}$
$[∵^{n} C_{r}^{n - 1} C_{r - 1}]$
$= (n + 1) {.2}^{n} - n \times 2^{n - 1}$
$= 2^{n - 1} . (2 (n + 1) - n)$
$= 2^{n - 1} (n + 2)$

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