Find the value of - r -0 n r n C rA- n -2n-1B- n-1 2n-1C- n-1 2nD- n - 2n

The correct option is A ^nC_r=n r×^n 1C_r 1 ⇒ ^r.nC_r = n × ^n 1C_r 1 ⇒∑_r=0^n r ^nC_r=∑_r=1^n n ^n 1C_r 1(∵^0.nC_0=0) =n∑_r=1^n ^n 1C_r 1 =(^n 1C_0+^n 1C_1+^ ...

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Byju's Answer

Standard XII

Mathematics

Binomial Coefficients

Find the valu...

Question

Find the value of $\sum_{r = 0}^{n} r^{n} C_{r}$

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Solution

The correct option is C

$^{n} C_{r} = \frac{n}{r} \times^{n - 1} C_{r - 1}$
$\Rightarrow^{r . n} C_{r} = n \times^{n - 1} C_{r - 1}$
$\Rightarrow \sum_{r = 0}^{n} r^{n} C_{r} = \sum_{r = 1}^{n} n^{n - 1} C_{r - 1} (∵^{0. n} C_{0} = 0)$
= $n \sum_{r = 1}^{n}^{n - 1} C_{r - 1}$
= $(^{n - 1} C_{0} +^{n - 1} C_{1} +^{n - 1} C_{1} +^{n - 1} C_{2} . . . . .^{n - 1} C_{n - 1})$
= $n {.2}^{n - 1}$

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