Q.Find the coefficient of $\dfrac{1}{x^{17}}$ in the expansion of $\left(x^4 - \dfrac{1}{x^3}\right)^{15}$.

The key idea here is the Binomial Theorem Expansion. We use the general term of the expansion to find the power of $x$ and then determine the corresponding coefficient.

The general term $T_{r+1}$ in the expansion of $(a+b)^n$ is given by:

For the given expression $\left(x^4 - \dfrac{1}{x^3}\right)^{15}$, we have $a = x^4$, $b = -x^{-3}$, and $n = 15$.

The general term is:

$T_{r+1} = \binom{15}{r} (x^4)^{15-r} (-x^{-3})^r$

$T_{r+1} = \binom{15}{r} x^{4(15-r)} (-1)^r (x^{-3})^r$

$T_{r+1} = \binom{15}{r} (-1)^r x^{60-4r} x^{-3r}$

$T_{r+1} = \binom{15}{r} (-1)^r x^{60-7r}$

We need the coefficient of $\dfrac{1}{x^{17}}$, which means the power of $x$ must be $-17$.

Equating the exponent of $x$ to $-17$:

$60 - 7r = -17$

$7r = 60 + 17$

$7r = 77$

$r = 11$

Now, substitute $r=11$ back into the coefficient part of the general term:

Coefficient $= \binom{15}{11} (-1)^{11}$ …

When expanding a binomial expression like $(a+b)^n$, we often don't need to write out all the terms. Instead, we can use the Binomial Theorem to find a specific term or its coefficient. The core idea is that each term in the expansion follows a predictable pattern.

Here, $a$ represents the first term of the binomial, $b$ represents the second term, and $n$ is the power to which the binomial is raised. By setting the power of $x$ in this general term equal to the desired power, we can find the specific value of $r$ that corresponds to the term we are looking for. Once $r$ is known, we can substitute it back into the formula to find the full coefficient.

Let's apply this to the given problem.

1. Identify $a$, $b$, and $n$ from the given expression.

   The expression is $\left(x^4 - \dfrac{1}{x^3}\right)^{15}$.

   Comparing this to $(a+b)^n$:

   • $a = x^4$
   • $b = -\dfrac{1}{x^3}$ (Note the negative sign is part of $b$)
   • $n = 15$

2. Write down the general term $T_{r+1}$ using these values.

   Using the formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$:

3. Simplify the general term to isolate the powers of $x$. We need to combine all the $x$ terms. Remember the exponent rules: $(x^m)^p = x^{mp}$ and $\dfrac{1}{x^k} = x^{-k}$.

Now, combine the $x$ terms using $x^m \cdot x^p = x^{m+p}$:

This is the simplified general term, showing the coefficient part and the $x$ part.

4. Equate the power of $x$ in the general term to the desired power.

We are looking for the coefficient of $\dfrac{1}{x^{17}}$. This can be written as $x^{-17}$.

So, we set the exponent of $x$ from our general term equal to $-17$:

5. Solve for $r$.

Since $r=11$ is an integer and $0 \le 11 \le 15$, this is a valid value for $r$.

6. Substitute the value of $r$ back into the coefficient part of the general term. …