For integers $n$ and $r$, let $\binom{n}{r}= \begin{cases}n C_{r}, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise }\end{cases}$ The maximum value of k for which the sum \[ \sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i}+\sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i} \text { exists, is equal to. } \]

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Binomial Theorem

For integers

n

and

r

, let

(r n) = {n C_{r}, 0, if n \geq r \geq 0 otherwise

The maximum value of k for which the sum

i = 0 \sum k (i 10) (k - i 15) + i = 0 \sum k + 1 (i 12) (k + 1 - i 13) exists, is equal to.

Answer: 10

For the first sum

\sum_{i = 0}^{k} (i 10) (k - i 15)

to exist, we need

0 \leq i \leq 10

and

0 \leq k - i \leq 15

. This gives

k \leq 15

and

k \leq 10 + 15 = 25

. The binding constraint is

k \leq 10

(since we need at least one valid term).

For the second sum

\sum_{i = 0}^{k + 1} (i 12) (k + 1 - i 13)

to exist, we need

0 \leq i \leq 12

and

0 \leq k + 1 - i \leq 13

. This gives

k + 1 \leq 13 + 12 = 25

.

The maximum value of

k

for which both sums exist is

k = 10

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Numerical answer