Binomial Theorem
Properties of Binomial Coefficients
IIT-JEE 2005
Grade 11
Question:
The sum $\sum_{r=0}^{10} \binom{10}{r} \binom{20}{m - r}$ (where $\binom{p}{q} = 0$ if $p < q$) is maximum when $m$ equals:
(1) $15$
(2) $10$
(3) $20$
(4) $30$
Step-by-Step Solution
Key Concept: Recognise this sum as the coefficient of $x^m$ in $(1+x)^{10}(1+x)^{20} = (1+x)^{30}$ by Vandermonde's identity, and recall which binomial coefficient of $(1 + x)^{30}$ is the largest.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)