Question 1080977
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nc1+2nc2+3nc3+...+nncn
~~~~~~~~~~~~~~~~~


<pre>
The sum is 

Sum = {{{C[n]^1}}} + {{{2*C[n]^2}}} + {{{3*C[n]^3}}} + . . . + {{{n*C[n]^n}}}.


The common term is

{{{k*C[n]^k}}} = {{{k*((n*(n-1)*(n-2)* ellipsis *(n-k+1))/(1*2*3* ellipsis *k))}}} = {{{cross(k)*((n*(n-1)*(n-2)* ellipsis *(n-k+1))/(1*2*3* ellipsis *cross(k)))}}} = {{{(n*(n-1)*(n-2)* ellipsis *(n-k+1)/(1*2*3* ellipsis *(k-1)))}}} = {{{n*C[n-1]^(n-1)}}}.

Therefore,

Sum = {{{C[n]^1}}} + {{{2*C[n]^2}}} + {{{3*C[n]^3}}} + . . . + {{{n*C[n]^n}}} = {{{n*C[n-1]^0}}} + {{{n*C[n-1]^1}}} + {{{n*C[n-1]^2}}} + . . . {{{n*C[n-1]^(n-1)}}} = {{{n*(C[n-1]^0 + C[n-1]^1 + C[n-1]^2 + ellipsis + C[n-1]^(n-1))}}}.

It is widely known (*) that 

{{{C[n-1]^0 + C[n-1]^1 + C[n-1]^2 + ellipsis + C[n-1]^(n-1)}}} = {{{2^(n-1)}}}.   (*)

Hence,

Sum = {{{n*2^(n-1)}}}.


Regarding the statement (*) see the lesson
    <A HREF=https://www.algebra.com/algebra/homework/Permutations/Remarkable-identities-for-Binomial-Coefficients.lesson>Remarkable identities for Binomial Coefficients</A> 
in this site. 
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Solved.