The coefficient of $x^{91}$ in the series $^{100}{C_1}\,{2^8}.\,{\left( {1\, – \,x} \right)^{99}}\, + {\,^{100}}{C_2}\,{2^7}.\,{\left( {1\, – \,x} \right)^{98}}\, + {\,^{100}}{C_3}\,{2^6}.\,{\left( {1\, – \,x} \right)^{97}}\, + \,….\, + {\,^{100}}{C_9}\,{\left( {1\, – \,x} \right)^{91}}$ is equal to –

Let ${s_1} = \mathop \sum \limits_{j = 1}^{10} j\left( {j – 1} \right)\left( {\begin{array}{*{20}{c}}{10}\\j\end{array}} \right)\;,$$\;{s_2} = \mathop \sum \limits_{j = 1}^{10} j\;\left( {\begin{array}{*{20}{c}}{10}\\j\end{array}} \right)\;and,$${s_3} = \mathop \sum \limits_{j = 1}^{10} {j^2}\left( {\begin{array}{*{20}{c}}{10}\\j\end{array}} \right)\;,\;$

Statement $-1$:${s_3} = 55 \times {2^9}$

Statement $-2$: ${s_1} = 90 \times {2^8}\;$ and ${s_2} = 10 \times {2^8}$ 

If number of terms in the expansion of ${(x – 2y + 3z)^n}$ are $45$, then $n=$

If ${(1 – x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + …. + {a_{2n}}{x^{2n}}$, then ${a_0} + {a_2} + {a_4} + …. + {a_{2n}} = $

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