If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) ...... (1 + x + x^2 + x^3 + ...... + x^n)$

$\equiv  a_0 + a_1x + a_2x^2 + a_3x^3 + ...... + a_mx^m$ then $\sum\limits_{r\, = \,0}^m {\,\,{a_r}}$ has the value equal to

In the expansion of ${(1 + x)^n}$ the sum of coefficients of odd powers of $x$ is

Let ${\left( {1 + x + {x^2}} \right)^{20}}\left( {2x + 1} \right) = {a_0} + {a_1}{x^1} + {a_2}{x^2} + ... + {a_{41}}{x^{41}}$ , then $\frac{{{a_0}}}{1} + \frac{{{a_1}}}{2} + .... + \frac{{{a_{41}}}}{{42}}$ is equal to 

A possible value of $^{\prime}x^{\prime}$, for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$ , is:

If $r,k,p \in W,$ then $\sum\limits_{r + k + p = 10} {{}^{30}{C_r} \cdot {}^{20}{C_k} \cdot {}^{10}{C_p}} $ is equal to -

The sum of coefficients in the expansion of ${(x + 2y + 3z)^8}$ is