Question

If \[ \binom{p}{q} = \binom{p}{q} \quad \text{and} \quad \sum_{i=0}^m \binom{10}{i} \binom{20}{m-i} \text{ is maximum, then find } m. \]

• A. 10
• B. 12
• C. 15
• D. 20

Hint:

Use the Vandermonde's identity to simplify and find maxima of binomial coefficient sums.

Answer 1

The Correct Option is C

The given sum is the convolution of binomial coefficients: \[ \sum_{i=0}^m \binom{10}{i} \binom{20}{m - i} = \binom{30}{m}. \] The maximum value of $\binom{30}{m}$ occurs at $m = \lfloor \frac{30}{2} \rfloor = 15$. Hence, $m = 15$.