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  4. 1. nC1+2. nC2+3. nC3+.....+n nCn is equal to

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Q. $1.\,{}^nC_1+2.\,{}^nC_2+3.\,{}^nC_3+.....+n\,{}^nC_n $ is equal to

Binomial Theorem

Solution:

Since $r\cdot^nC_r = n\cdot^{n-1}C_{r-1}$
$\therefore $ reqd. sum
$= n\left[^{n-1}C_{0}+\,{}^{n-1}C_{1}+\,{}^{n-1}C_{2} + .......... +\,{}^{n-1}C_{n-1}\right]$
$= n\cdot2^{n-1}$