If n is a positive integer and ${C_k} = {\,^n}{C_k}$, then the value of ${\sum\limits_{k = 1}^n {{k^3}\left( {\frac{{{C_k}}}{{{C_{k – 1}}}}} \right)} ^2}$ =

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 ${}^{21}{C_1} + 3.{}^{21}{C_3} + 5.{}^{21}{C_5} + ……19{}^{21}{C_{19}} + 21.{}^{21}{C_{21}} = k$ Then number of prime factors of $k$ is

Let $\left( a + bx + cx ^2\right)^{10}=\sum \limits_{ i =0}^{20} p _{ i } x ^{ i }, a , b , c \in N$. If $p _1=20$ and $p _2=210$, then $2( a + b + c )$ is equal to

The value of $^{4n}{C_0}{ + ^{4n}}{C_4}{ + ^{4n}}{C_8} + ….{ + ^{4n}}{C_{4n}}$ is