The sum Σᵢ₌₀ᵐ (10/i) (20/(m-i)), (where (p/q) = 0 if p

Q: Wait, why can't we just expand the sum manually? It looks like a simple series.

That is a common trap! While you could technically expand it for small $m$, the JEE Advanced level requires you to see the bigger picture. The summation is a general form, and manually calculating it for every $m$ would be incredibly time-consuming and prone to error. By recognizing it as a coefficient in a polynomial product, you bypass the tedious work and jump straight to the elegant solution using Vandermonde's Identity.

Q: How do I know when to use Vandermonde's Identity in an exam?

Look for the tell-tale signs: a sum of products of two binomial coefficients where the lower indices (like $i$ and $m-i$) add up to a constant ($m$). Whenever you see $\sum \binom{n}{i} \binom{k}{m-i}$, your brain should immediately scream 'Vandermonde's Identity!' It is a powerful tool that transforms a complex summation into a single, manageable binomial coefficient.

Q: Why is the maximum at $m=15$? Is it always $n/2$?

Great question! Yes, for any binomial coefficient $\binom{n}{r}$, the values increase until they reach the middle and then decrease. If $n$ is even, the peak is exactly at the middle, $r = n/2$. If $n$ were odd, there would be two middle terms, both giving the same maximum value. Since $n=30$ is even, the maximum occurs at $m = 30/2 = 15$. It is all about the beautiful symmetry of Pascal's triangle!

Question

The sum $\sum_{i=0}^m \binom{10}{i} \binom{20}{m-i}$, (where $\binom{p}{q} = 0$ if $p < q$) is maximum when $m$ is

Options:

(A) 5

(B) 10

(C) 15

(D) 20

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a massive, intimidating wall of a summation: 
$$S = \sum_{i=0}^m \binom{10}{i} \binom{20}{m-i}$$

At first glance, it looks like a nightmare of combinations. But in the world of JEE Advanced, we don't fight the wall; we find the door.

The door here is the **Polynomial Bridge**. Instead of treating this as a raw calculation, we view it through the lens of algebra.

Consider the two binomial expansions: 
$$(1+x)^{10} = \sum_{i=0}^{10} \binom{10}{i} x^i$$ 
and 
$$(1+x)^{20} = \sum_{j=0}^{20} \binom{20}{j} x^j$$

When we multiply these two, we are essentially looking for the coefficient of $x^m$ in the product $(1+x)^{10} \cdot (1+x)^{20}$.

### The Vandermonde Revelation

When you multiply $(1+x)^{10}$ by $(1+x)^{20}$, the laws of indices tell us that we simply add the exponents: 
$$(1+x)^{10} \cdot (1+x)^{20} = (1+x)^{30}$$

Now, the coefficient of $x^m$ in the expansion of $(1+x)^{30}$ is simply $\binom{30}{m}$.

This is the essence of **Vandermonde's Identity**: 
$$\sum_{i=0}^m \binom{n}{i} \binom{k}{m-i} = \binom{n+k}{m}$$

We have successfully condensed a complex, multi-term summation into a single, elegant binomial coefficient: 
$$S = \binom{30}{m}$$

### The Symmetry of the Peak

Now, we reach the final act of our journey. We need to maximize $S = \binom{30}{m}$.

Think of Pascal's triangle. The values in the $30^{th}$ row start small, grow steadily, reach a peak in the middle, and then mirror that growth on the way down.

This bell-like curve is the heart of binomial distribution. For any row $n$, the maximum value is found at the center.

Since $n=30$ is an even number, the center is exactly at: 
$$m = \frac{30}{2} = 15$$

There is no need for complex calculus or derivatives here; just the pure, geometric beauty of symmetry.

Our final answer is **$m=15$**.

You have just navigated one of the most fundamental identities in combinatorics. Keep this tool in your arsenal, and the next time you see a sum of products of binomial coefficients, you will know exactly how to bridge the gap.

Animated Solution for Mathematics - Binomial Theorem: The sum $\sum_{i = 0}^{m} (i 10) (m - i 20)$ , (where $(q p) = 0$ if $p < q$ ) is maximum when $m$ is

Select Answer:

Visualized Solution

The Given Summation

The Polynomial Connection

Setting up the Expansions

Forming $x^{m}$

Combining the Powers

Identifying the Coefficient

Property of Maximum Coefficient

Calculating the Maximum

Conclusion and Takeaway

The Sigma Insight: Properties of Binomial Coefficients

Analyzing the Setup

The Vandermonde Revelation

The Symmetry of the Peak

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