The sum  ${ }^{404}{C}_4{-}^4{C}_1{.}^{303}{C}_4{+}^4{C}_2{.}^{202}{C}_4{-}^4{C}_3{.}^{101}{C}_4$ is equals to  ${\left(101\right)}^k$ where k is ----

${ }^4{C_0}^{404}{C}_4{-}^4{C_1}^{303}{C}_4{+}^4{C_2}^{202}{C}_4{-}^4{C_3}^{101}{C}_4{+}^4{C}_4{-}^{4}C4$

= coefficient of  $x^4$  in  

 = coefficient of  $x^4$  in $\{{[{(1+x)}^{101}-1]}^4-1\}$

= coefficient of  $x^4$ in   ${[\cancel{1}+101x{+}^{101}{C}_2{x}^2+\mathrm{.....}+x^{101}\cancel{\cancel{-}}\cancel{1}]}^4-1$

= coefficient of  $x^4$  in  $\{x^4{[101{+}^{101}{C}_{2}x+\mathrm{.......}+x^{100}]}^4-1\}$

$$\begin{gathered} {=}^4{C}_0{\left(101\right)}^4={\left(101\right)}^4={\left(101\right)}^k \\ \therefore k=4 \end{gathered}$$