Question

Find the coefficient of $x^{50}$ in the expression:
${(1+x)}^{1000}+2x{(1+x)}^{999}+3x^2{(1+x)}^{998}+....+1001x^{1000}$

• A. ${{}^{1000}C}_{50}$
• B. ${{}^{1001}C}_{50}$
• C. ${{}^{1002}C}_{50}$
• D. ${{}^{1003}C}_{50}$

Answer 1

Answer: (C)

${{}^{1002}C}_{50}$

Let $S={(1+x)}^{1000}+2x{(1+x)}^{999}+3x^2{(1+x)}^{998}+3x^3{(1+x)}^{997}+....+1001x^{1000}$ .....(1)
$\frac{x}{(1+x)}S=x{(1+x)}^{999}+2x^2{(1+x)}^{998}+3x^3{(1+x)}^{997}+....+1000x^{1000}+\frac{1001}{(1+x)}{x}^{1001}$ ....(2)
On subtracting (2) from (1), we get

$(1-\frac{x}{1+x})S=(1+x{)}^{1000}+x{(1+x)}^{999}+x^2{(1+x)}^{998}+x^3{(1+x)}^{997}+....+x^{1000}-\frac{1001}{(1+x)}{x}^{1001}$
$\Rightarrow \left(\frac{1}{1+x}\right)S=(1+x{)}^{1000}+x{(1+x)}^{999}+x^2{(1+x)}^{998}+x^3{(1+x)}^{997}+....+x^{1000}-\frac{1001}{(1+x)}{x}^{1001}$
$\Rightarrow S=(1+x{)}^{1001}+x{(1+x)}^{1000}+x^2{(1+x)}^{999}+x^3{(1+x)}^{998}+....+x^{1000}(1+x)-1001x^{1001}$
$\Rightarrow S=\frac{(1+x{)}^{1001}[1-{\left(\frac{x}{1+x}\right)}^{1001}]}{1-\frac{x}{1+x}}-1001x^{1001}$
$\Rightarrow S=(1+x{)}^{1002}[1-{\left(\frac{x}{1+x}\right)}^{1001}]-1001x^{1001}$
$\Rightarrow S=(1+x{)}^{1002}-(1+x)x^{1001}-1001x^{1001}$
It is clear that in above sum , only first term can have $x^{50}$

So , coefficient of $x^{50}$ in $(1+x{)}^{1002}$ is ${}^{1002}{C}_{50}$