The sum ∑i=0m.  10Ci×20Cm−i where pCq=0 if p

We have, ∑i=0m 10Ci×20Cm−i= Number of ways of selecting m objects from two groups of objects one group containing 10 distinct objects of one type and another group containing 20 distinct objects of second kind.=30CmClearly,  30Cm is maximum when m=15, because  nCr is maximum for r=n/2 if n is even and when n is odd it ismaximum for r=n+12 or n−12.