Animated Solution for Mathematics - Binomial Theorem: If the number of terms in the expansion of $(1-\frac{2}{x}+\frac{4}{x^2}{)}^n,x\ne 0$, is 28, then the sum of the coefficients of all the terms in this expansion, is

Imagine you are standing before a complex expression: $(1-\frac{2}{x}+\frac{4}{x^2}{)}^n$. At first glance, it looks intimidating; it is not a simple binomial, but a trinomial.

In the world of JEE Advanced, we often encounter problems that test our ability to generalize. The binomial theorem is a powerful tool, but it is just a special case of the broader multinomial theorem.

To find $n$, we use the general formula for the number of terms in a multinomial expansion: ${}^{n+k-1}{C}_{k-1}$, where $k$ is the number of terms inside the bracket. Here, our $k$ is 3.

Substituting this into our formula, we get ${}^{n+3-1}{C}_{3-1}$, which simplifies to ${}^{n+2}{C}_2$. This is the key that unlocks the door.

Now, we set our formula equal to the given number of terms: ${}^{n+2}{C}_2=28$. Expanding this combination using factorials, we get:

With a little algebraic grace, the factorials simplify to:

Multiplying both sides by 2, we arrive at $(n+1)(n+2)=56$. We are looking for two consecutive integers whose product is 56.

A quick mental check reveals $7\times 8=56$. Thus, $n+1=7$, which means $n=6$. We have successfully navigated the first half of the problem!

Now for the final act: we need the sum of the coefficients. Many students might be tempted to expand the trinomial, but that is a trap designed to waste your precious time.

Remember the fundamental property of polynomials: to find the sum of all coefficients, simply set all variables to 1. Let $P(x)=(1-\frac{2}{x}+\frac{4}{x^2}{)}^6$.

The sum of the coefficients is simply $P(1)$. Substituting $x=1$ into our expression, we get:

This simplifies to $(1-2+4{)}^6$, which is $(3{)}^6$. Calculating $3^6$ is straightforward:

And there it is! Through logic and the power of substitution, we have arrived at the answer: 729.

Always remember, in JEE, the most elegant path is often the one that relies on fundamental properties rather than brute force. Keep practicing, and keep falling in love with the beauty of mathematics.