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The sum sum(l=0)^(m) ({:(10),(i):}) ({:...

The sum `sum_(l=0)^(m) ({:(10),(i):}) ({:(20),(m - i):})`, where `({:(p),(q):})` = 0 if `p gt q`, is maximum when m is equal to

A

5

B

10

C

15

D

20

Text Solution

Verified by Experts

`underset(i=0)overset(m)sum{:((10),(i)):}{:((20),(m-i)):}` is the coefficient of `x^(m)` in the expansion of `(1+x)^(10)(x+1)^(20),`
`impliesunderset(i=0)overset(m)sum{:((10),(i)):}{:((20),(m-i)):}` is the coefficient of `x^(m)` in the expansion of `(1+x)^(30)`
i.e. `underset(i=0)overset(m)sum{:((10),(i)):}{:((20),(m-i)):}=""^(30)C_(m)={:((30),(m)):}" "...(i)`
and we know that, `{:((n),(r)):}` is maximum, when
Hence, `{:((30),(m)):}` is maximum when m = 15.
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