Question

The maximum value of k for which the sum $\sum_{i=0}^k \binom{10}{i} \binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i} \binom{13}{k+1-i}$ exists, is equal to __________.

Hint:

Vandermonde's Identity $\sum \binom{r}{i} \binom{n}{k-i} = \binom{r+n}{k}$ is essential for sums involving products of binomial coefficients.

Answer 1

Correct Answer: 24

Step 1: By Vandermonde's Identity, $\sum_{i=0}^k \binom{n}{i}\binom{m}{k-i} = \binom{n+m}{k}$.
Step 2: First sum $= \binom{10+15}{k} = \binom{25}{k}$.
Step 3: Second sum $= \binom{12+13}{k+1} = \binom{25}{k+1}$.
Step 4: For the binomial coefficients to exist (be non-zero), $k \leq 25$ and $k+1 \leq 25$.
Step 5: Therefore, $k \leq 24$. Max value is 24.