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Q.

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

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a

maximum value of/(m) is  50C25

b

f(0)+f(1)+⋯+f(50)=250

c

f(m) is always, divisible by 50(1≤m≤49)

d

The value of ∑m=050 (f(m))2=100C50

answer is A.

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Detailed Solution

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