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The sum sum(i=0)^m ((10)c,(i))((20),(m-...

The sum `sum_(i=0)^m ((10)c,(i))((20),(m-1))`, where `((p),(q))=0` if `p lt q`, is maximum when m is equal to (A) 5 (B) 10 (C) 15 (D) 20

A

5

B

10

C

15

D

20

Text Solution

Verified by Experts

The correct Answer is:
c
`sum_(i=0)^m =((10),(i))^2((20),(m-i))=sum_(i=0)^m " "^(10)C_(i)""^(20)C_(m-i)`
`=^(10) C_(0) cdot ^(20)C_(m)+^(10)C_(1) cdot ^(20) C_(m-1)+...+^(10)C_(m) cdot ^20C_(0)`
= Coefficient of `x^(m)` in the expansion of product
`(1+x)^(10) (1+x)^(20)`
= Coefficient of `x^(m)` in the expansion of product `(1+x)^(30) = ^(30) C_(m)`
To get maximum value of the given sum `""^(30)C_(m)` shouuld be
maximum. Which is so, when `m=30/2=15`
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