Mathematics - Binomial Theorem Question with Solution | TestHub

${}^{50}{C}_6-{}^5{C}_1{}^{40}{C}_6+{}^5{C}_2{}^{30}{C}_6-{}^5{C}_3{}^{20}{C}_6+{}^5{C}_4{}^{10}{C}_6=$ coefficient of $x^6$ in $[{}^5{C}_0(1+x{)}^{50}-{}^5{C}_1(1+x{)}^{40}$ $+{}^5{C}_2(1+x{)}^{30}-{}^5{C}_3(1+x{)}^{20}+{}^5{C}_4(1+x{)}^{10}-{}^5{C}_5(1+x{)}^0]=$ coefficient $x^6$ in ${\left[(1+x{)}^{10}-1\right]}^5$ = coefficient of $x^6$ in ${\left({}^{10}{C}_{1}x+{}^{10}{C}_2{x}^2+\dots \right)}^5={}^5{C}_1\left({}^{10}{C}_2\right){\left({}^{10}{C}_1\right)}^4=2250000$.