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The value of sum(r=0)^(20)r(20-r)(^(20)C...

The value of `sum_(r=0)^(20)r(20-r)(^(20)C_r)^2` is equal to `400^(39)C_(20)` b. `400^(40)C_(19)` c. `400^(39)C_(19)` d. `400^(38)C_(20)`

A

` 400 ""^(37)C_(20)`

B

`400 ""^(40)C_(19)`

C

`400 ""^(38)C_(19)`

D

`400 ""^(38)C_(20)`

Text Solution

Verified by Experts

The correct Answer is:
d
`sum_(r=0)^(20) r(20-r)xx(""^(20)C_(r))^(2)`
` sum_(r=0)^(20)rxx ""^(20)C_(r) (20-r)xx""^(20)C_(20-r) = sum_(r=0)^(20) 20*""^(19)C_(r-1) xx20xx""^(19)C_(19-r)`
` = 400 sum_(r=0)^(20) ""^(19)C_(r-1)xx""^(19)C_(19-r)`
`= 400xx" Coefficient of " x^(18) "in" (1+x)^(19)(1 + x)^(19)`
` = 400xx""^(38)C_(18) = 400xx""^(38)C_(20)` .
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