Animated Solution for Mathematics - Binomial Theorem: The coefficient of $x^4$ in $(\frac{x}{2}-\frac{3}{x^2}{)}^{10}$ is

Imagine you are standing before a massive, daunting expression: $(\frac{x}{2}-\frac{3}{x^2}{)}^{10}$. If you were to expand this manually, you would be writing out eleven distinct terms, each more complex than the last.

As a JEE aspirant, you have a superpower: the Binomial Theorem. We do not need to see the whole forest; we only need to find one specific tree. Let us embark on this journey to find the coefficient of $x^4$.

The General Term

The Binomial Theorem provides us with a telescope, a way to look at any specific term without expanding the whole expression. We call this the General Term, denoted as:

Here, our $n$ is $10$, our first term $a$ is $\frac{x}{2}$, and our second term $b$ is $-\frac{3}{x^2}$. Notice how we treat the negative sign as part of the term $b$. This is the first step of our precision.

The Algebraic Separation

Now, we substitute our values into the formula:

This looks intimidating, but let us break it down. We need to separate the constants from the variables. We pull out the numbers: ${}^{10}{C}_r$, $(\frac{1}{2}{)}^{10-r}$, and $(-3{)}^r$.

What remains are the powers of $x$: $x^{10-r}$ in the numerator and $(x^2{)}^r$ in the denominator. Using the laws of exponents, $\frac{x^{10-r}}{x^{2r}}$ simplifies beautifully to $x^{10-3r}$. We have now reduced the entire expression to a single, manageable variable term.

The Detective Work

We are looking for the coefficient of $x^4$. This means the exponent of $x$ in our general term must be exactly $4$. We set up the equation:

Solving this is straightforward: $-3r=4-10$, which leads to $-3r=-6$, and finally, $r=2$. We have found our target! The term we are looking for is the third term ($T_{2+1}=T_3$).

The Final Calculation

Now that we know $r=2$, we plug it back into our constant expression:

Let us calculate this with care. ${}^{10}{C}_2$ is:

The term $(\frac{1}{2}{)}^8$ is $\frac{1}{256}$, and $(-3{)}^2$ is $9$. Multiplying these together:

The elegance of this result is satisfying, isn't it? We started with a complex binomial and, through the structured application of the Binomial Theorem, arrived at a precise fraction.

Remember, the key to JEE Advanced is not speed, but structure. Always isolate your variable, solve for $r$, and calculate with precision. You have mastered this concept today!