Question

The value of the summation \( \sum_{k=1}^{n} \sum_{r=0}^{k} r^{n} C_{r} \) is 

(1) \( n \cdot 2^{n} \) 

(2) \( n(n+1) 2^{n-2} \) 

(3) \( n(2)^{n-1} \) 

(4) \( 2^{n-1} \)

Answer 1

Correct option is (3) n(2)ⁿ⁻¹

\(\sum^n_{k =1}\) \(\sum^k_{r = 0} r^n C_r\) 

\(= \sum^n_{k=1}\left(0^nC_0 + 1^nC_1+ 2^nC_2+3^nC_3 +....+k^nC_k\right)\)

\(=\, ^nC_1 + 2^nC_2 + 3^nC_3+...+n^nC_n\)

\(= n(2)^{n -1}\) 

\(\because(1 +x)^n =\, ^nC_0 + \,^nC_1x + \,^nC_2x^2 + .....+^nC_nx^n\)

Differentiate both sides with respect to x

\(n( 1 +x)^{n -1} = \,^nC_1 + 2^n C_2 x + 3^nC_3x^2 +^{n.n}C_nx^{n-1}\)

Put x = 1

\(\therefore \,^nC_1 + 2^nC_1 + 3^nC_3+....+^{n.n}C_n = n.2^{n-1}\)