Question

Coefficient of $x^{50}$,

$(x>0)$, in $(1+x{)}^{1000}+2x(1+x{)}^{999}+3x^2(1+x{)}^{998}+\dots$ is

• A. ${}^{1000}{C}_{50}$
• B. ${}^{1001}{C}_{50}$
• C. ${}^{1002}{C}_{50}$
• D. ${}^{1002}{C}_{49}$

Answer 1

The correct option is C ${}^{1002}{C}_{50}$

${(1+x)}^{1000}+2x{(1+x)}^{999}+3x^2{(1+x)}^{998}...{\Rightarrow (1+x)}^{1000}(1+2\left(\frac{x}{x+1}\right)+3{\left(\frac{x}{x+1}\right)}^2+...)$
Considering $\left|\frac{x}{x+1}\right|<1$, we get
Given expression $=(1+x{)}^{1000}{(1-\frac{x}{x+1})}^{-2}={(1+x)}^{1002}$ Coefficient of $x^{50}$ is ${}^{1002}{C}_{50}$.