The coefficient of $x^{53 }$in $\sum _{m=0}^{100}{ }^{100}{C}_m(x-5{)}^{100-m}(4{)}^m$ is

We have

$\sum _{m=0}^{100}{ }^{100}{C}_m(x-5{)}^{100-m}(4{)}^m=(x-5+4{)}^{100}=(x-1{)}^{100}$

 $\therefore$  Coefficient of $x^{53}$is ${ }^{100}{C}_{53}(-1{)}^{47}={-}^{100}{C}_{53}$