SOLUTION: Evaluate 1.nC1+2.nC2+3.nC3+...........+n.nCn

Let S be this sum:  S =  +  +  + . . .  + .    (1)

Notice that S is the same as 

S =  +  +  +  + . . .  + .


Consider 2S and write it in this form 

2S =  +        +         +          + . . .  +  +

  +   +  +  + . . .   +      + .


Group the terms in pairs combining one term from the upper line and one term from the lower line. You will get

2S =  +  +  +  + . . . + .     (2)


Use the fact that   =   for any k,  0 <= k <= n.

So you can rewrite (2) in the form

2S =  +  +  +  + . . . +  = 

   = .


Now use well known fact that  

      = .

     Regarding this fact, see the proof in the lesson Remarkable identities for Binomial Coefficients in this site).


Then you finally get  

     2S = ,   and, hence,  S = .