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To find the coefficient of \( y^5 \) in the expansion of \( (4y - 3)^9 \), we can use the Binomial Theorem. The Binomial Theorem states:

\[
(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r
\]

For the given expression \( (4y - 3)^9 \), we identify:
- \( a = 4y \)
- \( b = -3 \)
- \( n = 9 \)

We are looking for the term where the power of \( y \) is 5. In the general term of the expansion:

\[
\text{Term}_r = \binom{9}{r} (4y)^{9-r} (-3)^r
\]

For this term to include \( y^5 \), we require:

\[
(4y)^{9-r} = 4^{9-r} y^{9-r}
\]

To get \( y^5 \), we set \( 9 - r = 5 \), which gives \( r = 4 \).

Now we substitute \( r = 4 \) into the general term:

\[
\text{Term}_4 = \binom{9}{4} (4y)^5 (-3)^4
\]

Simplifying:

\[
\text{Term}_4 = \binom{9}{4} \cdot 4^5 \cdot y^5 \cdot 81
\]

Calculating each part:

\[
\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
\]

\[
4^5 = 1024
\]

\[
81 = 3^4
\]

Thus:

\[
\text{Term}_4 = 126 \cdot 1024 \cdot 81 \cdot y^5
\]

\[
\text{Term}_4 = 1048576 \cdot y^5
\]

So, the coefficient of \( y^5 \) is \( 1048576 \).

Would you like to go over any details, or do you have any questions?

Here are five related questions to expand your understanding:

1. How does the sign of \( b \) affect the terms in the binomial expansion?
2. What is the general expression for finding the coefficient of a specific term in a binomial expansion?
3. How would the coefficient change if the exponent in the original expression was different?
4. Can you find the coefficient of \( y^4 \) in the expansion of \( (4y - 3)^9 \)?
5. How would the expansion differ if the binomial was \( (4y + 3)^9 \)?

**Tip:** When dealing with binomial expansions, it's helpful to identify patterns in the coefficients and exponents to simplify your calculations.

Find the coefficient of y^5 in the expansion of (4y - 3)^9.

Learn how to find the coefficient of y^5 in the expansion of (4y - 3)^9 using the Binomial Theorem. This detailed solution covers key algebraic concepts and the application of binomial expansion, suitable for high school students.

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