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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