Question

${}^{404}{C}_4-{}^4{C}_1{}^{303}{C}_4{+}^4{C}_2{}^{202}{C}_4{-}^4{C}_3{}^{101}{C}_4$ is equal to

• A. $(401{)}^4$
• B. $(101{)}^4$
• C. $(201{)}^4$

Answer 1

The correct option is B $(101{)}^4$

The given expression is the coefficient of $x^4$ in the expansion of
${}^4{C}_0(1+x{)}^{404}{-}^4{C}_1(1+x{)}^{303}{+}^4{C}_2(1+x{)}^{202}{-}^4{C}_3(1+x{)}^{101}{+}^4{C}_4$ as $C_4$ available in each term.
= coefficient of $x^4$ in $\left[\right(1+x{)}^{101}-1{]}^4$
= coefficient of $x^4$ in [$1{+}^{101}{C}_{1}x{+}^{101}{C}_2{x}^2+\dots ..-1{]}^4$
= coefficient of $x^4$ in ${[}^{101}{C}_{1}x{+}^{101}{C}_2{x}^2+\dots {]}^4$
=coefficient of $x^4$ in [${}^{101}{C}_1{x}^4+$higher powers of $x^4$]
$=(101{)}^4$