Animated Solution for Mathematics - Binomial Theorem: The coefficient of $x^{301}$ in $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\cdots +x^{500}$ is:

Imagine you are standing before a massive, intimidating expression: $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\cdots +x^{500}$. Many students see this and immediately reach for a brute-force expansion, but that is a path to exhaustion.

In the world of JEE Advanced, the key is not to fight the problem, but to observe it. Look at the structure; we have terms being added, and each term is a transformation of the previous one.

Specifically, if you divide any term by the one before it, you get the same ratio:

This is the heartbeat of a Geometric Progression.

Once we identify this as a Geometric Progression, the nightmare begins to dissolve. We have our first term $a=(1+x{)}^{500}$, our common ratio $r=\frac{x}{1+x}$, and we have carefully counted $n=501$ terms.

We invoke the sum formula for a finite geometric series:

Substituting our values, we get:

Now, watch the magic happen in the denominator. We have $1-\frac{x}{1+x}$. Finding a common denominator, this becomes $\frac{1+x-x}{1+x}$, which simplifies beautifully to $\frac{1}{1+x}$.

When we divide by this fraction, it is the same as multiplying by its reciprocal, $(1+x)$. So, our expression becomes:

This simplifies to:

Distributing the $(1+x{)}^{501}$ inside the bracket, we get:

We have reduced a massive, terrifying series into just two simple terms: $(1+x{)}^{501}-x^{501}$. The question asks for the coefficient of $x^{301}$.

In the expansion of $(1+x{)}^{501}$, the general term is given by ${}^{501}{C}_r{x}^r$. To get $x^{301}$, we set $r=301$, giving us the coefficient ${}^{501}{C}_{301}$.

The second part, $-x^{501}$, has no $x^{301}$ term, so it contributes nothing. Thus, our answer is ${}^{501}{C}_{301}$.

Finally, we apply the symmetry property ${}^n{C}_r{=}^n{C}_{n-r}$ to find that:

And there it is—the elegance of mathematics revealed. You didn't need to expand five hundred terms; you just needed to see the structure. The final answer is ${}^{501}{C}_{200}$.