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  4. The expression nCo+2 nC1 + 3 nC2+.....+(n+1)nCn is equal to

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Q. The expression $ ^nC_o+2 ^nC_1 + 3 ^nC_2+.....+(n+1)^nC_n $ is equal to

AMUAMU 2011Permutations and Combinations

Solution:

We know,
$( 1 + x)^n = \,{}^nC_0 + \,{}^nC_1\, x + \,{}^nC_2\,x^2$
$ + ... + \,{}^nC_n\,x^n$
$\Rightarrow x(1+x)^n = \,{}^nC_0\,x + \,{}^nC_1\,x^2 + \,{}^n C_2\,x^3 + ...$
$ + \,{}^nC_n\, x^{n+1}$
On differentiating w.r.t. $x$, we get
$( 1 + x )^n + nx (1 + x)^{n-1} = \,{}^nC_0 + 2\cdot \,{}^nC_1\,x$
$ + 3 \cdot \,{}^nC_2\, x^2 + ... + (n + 1) \cdot ^nC_n \,x^n$
Put $ x = 1$, we get
$2^n + 2^{n-1}\cdot n = \,{}^nC_1 + 2 \cdot \,{}^nC_1 + 3 \cdot \,{}^nC_2$
$ + ... + (n + 1)\cdot ^nC_n$
$ = ( n + 2)2^{n-1}$