The sum of the series ` sum_(r=0) ^(n) ""^(2n)C_(r), ` is

Options

A. `2^(2n)`
B. `2^(n)`
C. `2^(2n) + ""^(2n)C_(n)`
D. `(1)/(2) (2^(2n) + ""^(2n)C_(n))`

Answer

D

Explanation

We have,
`sum_(r=0)^(n) ""^(2n)C_(r) = (1)/(2)sum_(r=0)^(n) (2^(2n) + ""^(2n)C_(n))` ltbbrgt `rArr sum_(r=0)^(n) ""^(2n)C_(r) = (1)/(2)sum_(r=0)^(n) ""^(2n)C_(2n) + ""^(2n)C_(n)r = (1)/(2)[ {sum_(r=0)^(2n) ""^(2n)C_(r)}+""^(2n)C_(n)r]`
`rArr sum_(r=0)^(n) ""^(2n)C_(r) = (1)/(2) (2^(2n) + ""^(2n) C_(n))` .