Question

The coefficient of `x^(301` ub the expansion of `(1+x)^(500)+x(1+x)^(499)+x^(2)(1+x)^(498)+…+x^(500)` is
A. `"^(501)C_(301)`
B. `"^(500)C_(301)`
C. `"^(501)C_(300)`
D. none of these

Answer 1

Correct Answer - A
`(a)` The given series
`S=(1+x)^(500)[1+(x)/(1+x)+((x)/(1+x))^(2)+....+((x)/(1+x))^(500)]`
`=(1+x)^(500)xx(1-((x)/(1+x))^(501))/(1-(x)/(1+x))`
`=(1+x)^(501)-x^(501)`
Hence, the coefficient of `x^(301)` in `S=^(501)C_(301)`.