Animated Solution for Mathematics - Binomial Theorem: The value ${\sum }_{r=0}^{22}{}^{22}{C}_r\cdot {}^{23}{C}_r$ is

Imagine you are standing before a complex summation:

At first glance, it looks like a daunting product of two binomial coefficients. Many students see this and immediately try to expand the factorials, leading to a messy, algebraic nightmare.

But stop. Take a breath. In the world of JEE Advanced, we don't brute-force; we look for the hidden structure. The variable $r$ in the lower indices is the obstacle preventing us from using the elegant tools in our arsenal.

To unlock this, we need to look at the symmetry property of binomial coefficients:

This is not just a formula; it is a profound geometric truth. Choosing $r$ items to include is exactly the same as choosing $n-r$ items to exclude.

Let us apply this to the second term, ${}^{23}{C}_r$. By replacing $r$ with $23-r$, we transform our expression into:

Notice what has happened. We have created a beautiful balance. The sum of the lower indices is now $r+(23-r)=23$. This is a constant, and the variable $r$ has been neutralized.

Now that we have a constant sum of lower indices, we have arrived at the gates of Vandermonde's Identity. This identity states that:

It is a powerful shortcut that allows us to collapse a product of combinations into a single, elegant term. Here, our $m$ is $22$, our $n$ is $23$, and our constant sum $p$ is $23$. The expression fits perfectly.

You might worry about the summation limit. Our sum goes to $22$, but the identity expects $p=23$. Do not panic.

If we were to include the $r=23$ term, we would be adding ${}^{22}{C}_{23}\cdot {}^{23}{C}_0$. Since ${}^{22}{C}_{23}=0$, we are adding nothing. The identity holds firm.

We simply add the upper indices: $22+23=45$. The lower index remains our constant sum, $23$. The entire expression simplifies to:

It is a moment of pure mathematical satisfaction when the complexity vanishes, leaving behind a single, clean result. You have mastered the symmetry, recognized the identity, and navigated the trap. The final answer is ${}^{45}{C}_{23}$.