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

Options

A. 5
B. 10
C. 15
D. 20

Explanation

We have, `sum_(i=0)^(m)""^(10)C_(i)xx""^(20)C_(m-i)`
= Number of ways of selecting m objects from two groups of objects one group containning 10 distinct objects of one type and another group containing 20 distinct objects of second kind.
`=""^(30)C_(m)`
Clearly, `=""^(30)C_(m)` is maximum when m=15, because `""^(n)C_(r)` is maximum for `r=n//2` if n is even and when n is odd it is maximum for r `r=(n+1)/(2)or(n-1)/(2)`.