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AIEEE 2002

JEE Main 2002 Mathematics question on Binomial Theorem

The sum ∑i=0m(10i)(20m−i)\displaystyle\sum_{i=0}^{m} \binom{10}{i}\binom{20}{m-i}, where (pq)=0\binom{p}{q} = 0 if p>qp > q, is maximum when mm is equal to
  1. A.
    5
  2. B.
    10
  3. C.
    15
  4. D.
    20
Show answer and solution

Correct option: C

Solution

The sum equals the coefficient of xm in (1+x)10 · (1+x)20 = (1+x)30, which is C(30, m). C(30, m) is maximized when m = 15 (the middle term of the expansion). Since m ranges from 0 to 20 (due to the constraint on i), m = 15 is valid and gives the maximum value C(30, 15).

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