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To expand \( (1 + 4y)^5 \) using Pascal's Triangle, let's follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients using Pascal's Triangle for the 5th power.**
   For the expansion of \( (a + b)^5 \), the coefficients from Pascal's Triangle are:
   \[
   1, 5, 10, 10, 5, 1
   \]

2. **Write out the binomial expansion using these coefficients.**
   The formula for binomial expansion is:
   \[
   (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
   \]

   For \( (1 + 4y)^5 \), \( a = 1 \) and \( b = 4y \), and we get:
   \[
   (1 + 4y)^5 = 1 \cdot (1)^5 + 5 \cdot (1)^4 \cdot (4y) + 10 \cdot (1)^3 \cdot (4y)^2 + 10 \cdot (1)^2 \cdot (4y)^3 + 5 \cdot (1) \cdot (4y)^4 + (4y)^5
   \]

3. **Calculate each term.**
   - **First term:** \( 1 \cdot (1)^5 = 1 \)
   - **Second term:** \( 5 \cdot (1)^4 \cdot (4y) = 5 \cdot 4y = 20y \)
   - **Third term:** \( 10 \cdot (1)^3 \cdot (4y)^2 = 10 \cdot 16y^2 = 160y^2 \)
   - **Fourth term:** \( 10 \cdot (1)^2 \cdot (4y)^3 = 10 \cdot 64y^3 = 640y^3 \)
   - **Fifth term:** \( 5 \cdot (1) \cdot (4y)^4 = 5 \cdot 256y^4 = 1280y^4 \)
   - **Sixth term:** \( (4y)^5 = 1024y^5 \)

4. **Combine all terms.**
   So, the expanded form is:
   \[
   (1 + 4y)^5 = 1 + 20y + 160y^2 + 640y^3 + 1280y^4 + 1024y^5
   \]

### Final Answer:
\[
1 + 20y + 160y^2 + 640y^3 + 1280y^4 + 1024y^5
\]

Would you like a detailed explanation on how Pascal's Triangle works or have any other questions?

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### Follow-Up Questions

1. How does Pascal's Triangle relate to the coefficients in binomial expansion?
2. Can you expand \( (1 + 3y)^4 \) using a similar method?
3. How would the expansion differ if the power was 6 instead of 5?
4. What happens if we expand a binomial where both terms have variables, like \( (x + y)^5 \)?
5. Can you find a way to derive the coefficients without using Pascal's Triangle?

### Tip
When expanding binomials with Pascal's Triangle, use each row corresponding to the power of the binomial for quick and accurate coefficient selection.

Use Pascal's Triangle to expand (1 + 4y)^5. Express your answer in simplest form.

Learn how to expand (1 + 4y)^5 using Pascal's Triangle. This example covers the binomial theorem and shows how to use Pascal's Triangle for quick coefficient selection. Ideal for students in Grades 9-11.

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