Question

The coefficient of $x^{50}$ in the expansion of ${\left(1+x\right)}^{100}+2x{\left(1+x\right)}^{99}+3x^2{\left(1+x\right)}^{98}+...+101x^{100},$ is

• A. ${}^{100}{C}_{50}$
• B. ${}^{101}{C}_{50}$
• C. ${}^{102}{C}_{50}$
• D. ${}^{103}{C}_{50}$

Answer 1

Answer: (C)

Let $S=(1+x{)}^{100}+2x(1+x{)}^{99}+3x^2(1+x{)}^{98}$
$+\dots +101x^{100}$
$\frac{x}{1+x}S=$
$x(1+x{)}^{99}+2x^2(1+x{)}^{98}+\dots +100x^{100}+101\frac{x^{101}}{1+x}$
$\Rightarrow \frac{S}{1+x}=(1+x{)}^{100}+x(1+x{)}^{99}+x^2(1+x{)}^{98}$
$+\dots +x^{100}-101\frac{x^{101}}{1+x}$
$\Rightarrow S=(1+x{)}^{102}-x^{102}-102x^{101}$
So, coefficient of $x^{50}$ in the expansion of $S={}^{102}{C}_{50}$