The value of the sum ${C_1} + 2{C_2} + 3{C_3} + 4{C_4} + .... + n{C_n}$ is equal to:

If $p$ and $q$ are positive integers,then the coefficients of $x^p$ and $x^q$ in the expansion of $(1 + x)^{p + q}$ are

If $1^2 \cdot \binom{15}{1} + 2^2 \cdot \binom{15}{2} + 3^2 \cdot \binom{15}{3} + \ldots + 15^2 \cdot \binom{15}{15} = 2^m \cdot 3^n \cdot 5^k$,where $m, n, k \in N$,then $m + n + k$ is equal to :-

If $C_j = {}^{n}C_j$,then $C_0 C_r + C_1 C_{r+1} + C_2 C_{r+2} + \ldots + C_{n-r} C_n = $

If $(1 + x + x^2)^n = a_0 + a_1x + a_2x^2 + \dots + a_{2n}x^{2n}$,then $a_0 + a_3 + a_6 + \dots =$

If $(1 + x)^n = C_0 + C_1x + C_2x^2 + .... + C_nx^n$,then $C_0C_2 + C_1C_3 + C_2C_4 + .... + C_{n-2}C_n$ equals