The value of `((100),(0))((200),(150))+((100),(1))((200),(151))+......+((100),(50))((200),(200))` equals (where `((n),(r ))="^(n)C_(r)`)

Options

A. `((300),(50))`
B. `((100),(50))((200),(150))`
C. `((100),(50))^(2)`
D. `((300),(50))^(2)`

Answer

A

Explanation

`(a)` Writing `((200),(r ))=((200),(200-r))`, we have
`((100),(0))((200),(50))+((100),(1))((200),(49))+((100),(2))((200),(48))+......+((100),(50))((200),(0))`
`=` Coefficient of `x^(50)` in the expansion of `(1+x)^(100)(x+1)^(200)`
`=` Coefficient of `x^(50)` in the binomial expansion of `(1+x)^(300)`
`=((300),(50))`