Question

The value of ${}^{404}{C}_4$ - ${}^4{C}_1$${}^{303}{C}_4$ + ${}^{202}{C}_4$ - ${}^4{C}_3$${}^{101}{C}_4$ =

• A. (404)⁴
• B. (101)⁴
• C. 0
• D. 1

Answer 1

The correct option is B

(101)⁴

The given expression is the coefficient of $x^4$ in

${}^4{C}_0(1+x{)}^{404}$ - ${}^4{C}_1(1+x{)}^{303}$ + ${}^4{C}_2(1+x{)}^{202}$ - ${}^4{C}_3(1+x{)}^{101}$ + ${}^4{C}_4$

i.e. in $\left[\right(1+x{)}^{101}-1{]}^4$

i.e. in $({}^{101}{C}_{1}x$ + ${}^{101}{C}_2{x}^2$ + ........ + ${}^{101}{C}_{101}{x}^{101}{)}^4$ = $(101{)}^4$

$\therefore$ option (b) is correct.