Define f(m) = sum from i = 0 to m of C(30, 30-i) * C(20, m-i), where C(p,q) denotes p-choose-q. Which of the following statements are correct?

Question

Accepted Answer

The maximum value of f(m) is C(50, 25). f(m) = sum_(i=0)^(m) C(30, 30-i)*C(20, m-i) = sum_(i=0)^(m) C(30, i)*C(20, m-i) = C(50, m) by Vandermonde's identity. So D is true. Maximum of C(50,m) is at m=25, giving C(50,25) — so A is true. Sum over m=0 to 50 of C(50,m) = 2⁵⁰ — so B is true. For C, C(50,m) divisible by 50 for 1<=m<=49 requires checking; it is not always true (e.g. C(50,1)=50 is divisible, but the claim needs careful verification — in fact for prime factors this does not hold universally). The correct options are A, B, and D.

Define f(m) = sum from i = 0 to m of C(30, 30-i) * C(20, m-i), where C(p,q) denotes p-choose-q. Which of the following statements are correct?

Solution

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