MAH-CET 2026 — Mathematics PYQ

Step 1: Simplify the given summation

The given series is:

$$S=\sum _{i=0}^m{}^{10}{C}_i\cdot {}^{20}{C}_{m-i}$$

This expression represents a classic combinatorial identity known as Vandermonde's Identity.

Alternatively, we can understand this by looking at the coefficients in a binomial expansion. The product ${}^{10}{C}_i\cdot {}^{20}{C}_{m-i}$ represents choosing $i$ items from a group of 10, and the remaining $(m-i)$ items from a separate group of 20.

Summing this from $i=0$ to $m$ is equivalent to finding the total number of ways to select $m$ items from a combined pool of $10+20=30$ items.

Therefore:

$$S={}^{30}{C}_m$$

Step 2: Condition for Maximum Value

We need to find the value of $m$ for which ${}^{30}{C}_m$ is maximized.

For any binomial coefficient ${}^n{C}_r$, the value is maximum at the middle term:

• If $n$ is even, the maximum value occurs at $r=\frac{n}{2}$.

• If $n$ is odd, the maximum value occurs at $r=\frac{n-1}{2}$ and $r=\frac{n+1}{2}$.

Here, $n=30$, which is an even number.

Thus, ${}^{30}{C}_m$ will be maximum when:

$$m=\frac{30}{2}=15$$

Conclusion

The sum is maximum when $m=15$.

Correct Option: (c) 15