The value of `sum_(k=0)^(7)[(((7),(k)))/(((14),(k)))sum_(r=k)^(14)((r ),(k))((14),(r ))]`, where `((n),(r ))` denotes `"^(n)C_(r )` is

Options

A. `6^(7)`
B. greater than `7^(6)`
C. `8^(7)`
D. greater than `7^(8)`

Answer

A, B

Explanation

`(a,b)` `sum_(k=0)^(7)(('^(7)C_(k))/('^(14)C_(k))sum_(r=k)^(14)'^(r )C_(k)*^(14)C_(r ))`
`=sum_(k=0)^(7)(('^(7)C_(k))/(14!)xxk!(14-k)!sum_(r=k)^(14)(r !)/(k!(r-k)!)*(14!)/(r!(14-r)!))`
`=sum_(k=0)^(7)('^(7)C_(k)sum_(r=k)^(14)'^(14-k)C_(r-k))`
`=sum_(k=0)^(7)'^(7)C_(k)*2^(14-k)=2^(14)sum_(k=0)^(7)'^(7)C_(k)((1)/(2))^(k)`
`=2^(14)*(1+(1)/(2))^(7)=6^(7) gt 7^(6)`