Animated Solution for Mathematics - Binomial Theorem: The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+...+x^{10}$ is :

The given expression is $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\cdots +x^{10}$. At first glance, it appears complex, but we can observe a distinct pattern.

The power of $(1+x)$ decreases from $10$ to $0$, while the power of $x$ increases from $0$ to $10$. This structure is the unmistakable heartbeat of a Geometric Progression (GP).

To solve this, we must identify the parameters of our GP. The first term, $a$, is $(1+x{)}^{10}$.

To find the common ratio, $r$, we divide the second term by the first:

Counting the terms from $x^0$ to $x^{10}$ gives us exactly $n=11$ terms. We now invoke the standard sum formula for a GP:

Substituting our values into the formula, we get:

Focusing on the denominator, $1-\frac{x}{1+x}$, we find a common denominator to simplify it:

Now, we multiply the term outside, $(1+x{)}^{10}$, by the reciprocal of the denominator, $(1+x)$, which yields $(1+x{)}^{11}$. When this $(1+x{)}^{11}$ multiplies the terms inside the bracket, the denominator $(1+x{)}^{11}$ cancels out perfectly.

The entire series collapses into a simple binomial expression:

Our goal is to find the coefficient of $x^7$ in $(1+x{)}^{11}-x^{11}$. Since $x^{11}$ contains no $x^7$ term, we only need to consider the expansion of $(1+x{)}^{11}$.

Using the Binomial Theorem, the coefficient of $x^r$ in $(1+x{)}^n$ is given by ${}^n{C}_r$. Here, $n=11$ and $r=7$.

We need to calculate ${}^{11}{C}_7$. Using the symmetry property of combinations, ${}^n{C}_r={}^n{C}_{n-r}$, we simplify this to ${}^{11}{C}_4$:

The final answer is 330. This problem demonstrates that even complex expressions often hide a simple, elegant structure.