Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 1: JEE Advanced Paper 1 - 2013

For non-negative integers $s$ and $r$, let $\left(\begin{array}{l}s\\ r\end{array}\right)=\left\{\begin{array}{cc}\frac{s!}{r!\left(s-r\right)!}& \mathrm{if} r\le s,\\ 0& \mathrm{if} r>s.\end{array}\right.$

For positive integers $m$ and $n$, let $g\left(m,n\right)=\sum _{p=0}^{m+n}\frac{f\left(m,n,p\right)}{\left(\begin{array}{c}n+p\\ p\end{array}\right)}$

Where for any non-negative integer $p$, $f\left(m,n,p\right)=\sum _{i=0}^p\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n+i\\ p\end{array}\right)\left(\begin{array}{c}p+n\\ p-i\end{array}\right)$

Then which of the following statement is/are TRUE?

Let $X={\left({}^{10}{C}_1\right)}^2+2{\left({}^{10}{C}_2\right)}^2+3{\left({}^{10}{C}_3\right)}^2+\dots +10{\left({}^{10}{C}_{10}\right)}^2$ , where ${}^{10}{C}_r, r\in \left\{\mathrm{1,2},\dots .,10\right\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is ______ .

Coefficient of $x^{11}$ in the expansion of ${\left(1+x^2\right)}^4 {\left(1+x^3\right)}^7 {\left(1+x^4\right)}^{12}$ is