if `sum_(r=0)^(25).^(50)C_(r)(.^(50-r)C_(25-r))=k(.^(50)C_25)`, then k equals: (a) `2^(25)` (b) `2^(25)-1` (c) `2^(24)` (d) `25^(2)`

Options

A. `2^(24)`
B. `2^(25)-1`
C. `(25)^(2)`
D. `2^(25)`

Answer

D

Explanation

`sum_(r=0)^(25)""^(50)C_(r).""^((50-r))C_(25-r)`
`sum_(r=0)^(25) (50!)/(r!(50-r)!)xx((50-r)!)/(25! xx (25-r)!)`
Now multiply and divide by 25!
`=sum_(r=0)^(25) (50!)/(r!) xx (25!)/(25!^(2).(25-r)!) = sum_(r=0)^(25) (50!)/(25! .25!)xx (25!)/(r!(25-r)!)=sum_(r=0)^(25)""^(50)C_(25).""^(25)C_(r)`
`""^(50)C_(25)sum_(r=0)^(25) ""^(25)C_(r)= ""^(50)C_(25).2^(25)=""^(50)C_(25).k`
Hence `k=2^(25)`