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If sum(r = 0)^(25) {""^(50)C(r)* ""^(50-...

If `sum_(r = 0)^(25) {""^(50)C_(r)* ""^(50-r)C_(25 - r)} = K(""^(50)C_(25)),`
then, K is equal to

A

`2^(24)`

B

`2^(25) - 1`

C

`2^(25)`

D

`(25)^(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given, `overset(25)underset(r = 0)sum {""^(50)C_(r)*""^(50-r)C_(25 - r)} = K " "^(50)C_(25)`
`rArr overset(25)underset(r = 0)sum ((50!)/(r!(50-r)!) xx ((50 - r)!)/((25 - r)!25!)) = K " "^(50)C_(25)`
`rArr" "overset(25)underset(r = 0)sum ((50!)/(25!25!) xx (25!)/(r!(25 - r)!)) = K" "^(50)C_(25)` [on multiplying `25!` in numerator and denominator.]
`rArr" "^(50)C_(25)overset(25)underset(r = 0)sum ""^(25)C_(r) = K" "^(50)C_(25)" " [because ""^(50)C_(25) = (50!)/(25!25!)]`
`rArr" "K = overset(25)underset(r = 0)sum ""^(25)C_(r) = 2^(25)`
`[because ""^(n)C_(0) + ""^(n)C_(1) + ""^(n)C_(2) + .....+ ""^(n)C_(n) = 2^(n)]`
`rArr" " K = 2^(25)`
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