Question

Find the coefficient of $x^{25}$ in expansion of expression $\sum _{r=0}^{50}{{}^{50}C}_r(2x-3{)}^r(2-x{)}^{50-r}$.

Answer 1

Now

$\sum _{r=0}^{50}{{}^{50}C}_r(2x-3{)}^r(2-x{)}^{50-r}=\left\{\right(2x-3)+(2-x){\}}^{50}=(x-1{)}^{50}$.

From this expression it is clear that the co-efficient of $x^{25}$ is ${}^{50}{C}_{25}(-1{)}^{25}={-}^{50}{C}_{25}$.