If f(m)=∑i=0m 3030−i20m−i where pq=pCq then

f(m)=∑i=0m 3030−i20m−i=∑i=0m 30i20m−i=50Cmf(m) is greatest when m = 25. Alsof(0)+f(1)+…+f(50)=50C0+50C1+50C2+…+50C50=250Also,  50Cm is not divisible by 50 for any m as 50 is not a prime number∑m=050 (f(m))2= 50C02+ 50C12+ 50C22+…+ 50C502=100C50