Question

The value of $\sum _{r=0}^3{}^8{C}_r\left({}^5{C}_{r+1}-{}^4{C}_r\right)$ is____

Answer 1

Answer: 220

$\sum _{r=0}^3{}^8{C}_r\left({}^5{C}_{r+1}-{}^4{C}_r\right)=\sum _{r=0}^3{}^8{C}_r{}^4{C}_{r+1}$
$={}^8{C}_0\times {}^4{C}_1+{}^8{C}_1\times {}^4{C}_2+{}^8{C}_2\times {}^4{C}_3+{}^8{C}_3\times {}^4{C}_4$
$=$ coefficient of $x^3$ in $(1+x{)}^4(1+x{)}^8$
$=$ coefficient of $x^3$ in $(1+x{)}^{12}$
$={}^{12}{C}_3=220$