The value of `(sumsum)_(0leilejlen) (""^(n)C_(i) + ""^(n)C_(j))` is equal to

Options

A. `n2^(n)`
B. `(n+1)2^(n)`
C. `(n +2) 2^(n)`
D. none of these

Answer

c

Explanation

We have,
`underset(0leilejlen)(sumsum)(""^(n)C_(i) + ""^(n)C_(j))` ltbrge `underset(0leilejlen)(sumsum)""^(n)C_(i) +underset(0leilejlen)(sumsum) ""^(n)C_(j)`
= `sum_(i=0)^(n) (n - i +1)""^(n)C_(i) + sum_(i=0)^(n) (j +1) ""^(n)C_(j)`
= `sum_(i=0)^(n) (n -r+1)""^(n)C_(r)+ sum_(i=0)^(n) (r +1)""^(n)C_(r)`
`sum_(i=0)^(n) (n +2)""^(n)C_(r)= (n+2) sum_(i=0)^(n) ""^(n)C_(j)= (n+2)2^(n)`.