The coefficient of `x^100` in the expansion `sum_(r=0) ^ 200 ( 1+x)^r` is

Options

A. `{:((200),(100)):}`
B. `{:((201),(102)):}`
C. `{:((200),(101)):}`
D. `{:((201),(101)):}`

Answer

D

Explanation

We have,
n`sum_(r=0)^(200) (1 + x)^(r) = 1 +(1 + x) + (1 + x)^(2) +… + (1 + x)^(200)`
`rArr sum_(r=0)^(200) (1 + x)^(r) = { ((1 + x)^(201) -1)/((1 + x) -1)} = (1)/(x) { (1 + x)^(201) -1}`
` therefore` Coefficient of `x^(10)` in `sum_(r=0)^(200) (1 + x)^(r)`
Coefficient of `x^(10)` in `(1 + x)^(201) = {:(20),(101):}`