Animated Solution for Mathematics - Binomial Theorem: If the constant term in the expansion of ${\left(\frac{\sqrt[5]{3}}{x}+\frac{2x}{\sqrt[3]{5}}\right)}^{12},x\ne 0$, is $\alpha \times 2^{8}imes\sqrt[5]{3}$, then $25\alpha$ is equal to :

Welcome, fellow explorer of the mathematical universe! Today, we are going to tackle a classic JEE Advanced problem. It is not just about crunching numbers; it is about seeing the hidden structure within a binomial expansion.

Imagine you are standing before the expression:

It looks intimidating, but remember, every complex expression is just a collection of simpler parts waiting to be organized.

In the world of binomial theorem, the General Term is our master key. It is the DNA of the entire expansion. The formula is:

Here, our $n$ is $12$, our $A$ is $\frac{3^{1/5}}{x}$, and our $B$ is $\frac{2x}{5^{1/3}}$. When we plug these into our formula, we get:

This is the blueprint. Now, we just need to find the right $r$ to unlock the constant term.

What is a 'constant term'? It is a term where the variable $x$ has completely vanished—or, more mathematically, where the power of $x$ is zero.

Let's isolate the $x$ parts. From the first term, we have $x^{-(12-r)}$, and from the second, we have $x^r$. When we multiply these, we add the exponents:

To make this term constant, we set the exponent to zero: $2r-12=0$, which gives us $r=6$. We have found our target!

Now that we know $r=6$, we substitute it back into our general term. This gives us:

Simplifying this, we get:

Don't let the fractional powers scare you. We can rewrite $3^{6/5}$ as $3^1\cdot 3^{1/5}$.

The value of ${}^{12}{C}_6$ is $924$. If we factor $924$, we get $231\times 4$, or $231\times 2^2$. Putting it all together:

The problem states the constant term is $\alpha \times 2^8\times \sqrt[5]{3}$. Comparing this to our result, we see that:

The question asks for $25\alpha$. Multiplying $\frac{693}{25}$ by $25$ gives us exactly $693$.

And there it is! The elegance of the cancellation is the reward for your patience. Keep practicing, and you will find that these problems are not obstacles, but stepping stones to mastery.