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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 ?

A

`400.^(39)C_(20)`

B

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

C

`400.^(39)C_(19)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
D
`=underset(r=0)overset(20)sumr(20-r)xx(.^(20)C_(r))^(2)=underset(r=0)overset(20)sumrxx.^(20)C_(r)(20-r)xx.^(20)C_(20-r)`
or `underset(r=0)overset(20)sum20.^(19)C_(r-1)xx20xx.^(19)C_(19-r)`
`= 400underset(r=0)overset(20)sum.^(19)C_(r-1)xx.^(19)C_(19-r)`
`= 400 xx "coefficient of" x^(19) "in" (1+x)^(19)(1+x)^(19)`
`= 400 xx .^(38)C_(18)`
`= 400 xx .^(38)C_(20)`
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