To find the first three terms of the binomial expansion of \((2p - y)^{52}\), we use the **Binomial Theorem**. The Binomial Theorem states that:

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

For the binomial \((2p - y)^{52}\), we have \(a = 2p\), \(b = -y\), and \(n = 52\).

The general term in the expansion is given by:

\[
T_{k+1} = \binom{n}{k} (2p)^{n-k} (-y)^k
\]

where \(T_{k+1}\) is the \((k+1)\)-th term, \(\binom{n}{k}\) is the binomial coefficient, and \(k = 0, 1, 2, \ldots, n\).

### First 3 Terms of the Expansion:

1. **First term (\(T_1\))**: \(k = 0\)

\[
T_1 = \binom{52}{0} (2p)^{52} (-y)^0 = 1 \cdot (2p)^{52} \cdot 1 = (2p)^{52}
\]

2. **Second term (\(T_2\))**: \(k = 1\)

\[
T_2 = \binom{52}{1} (2p)^{51} (-y)^1 = 52 \cdot (2p)^{51} \cdot (-y) = -52 \cdot (2p)^{51} \cdot y
\]

3. **Third term (\(T_3\))**: \(k = 2\)

\[
T_3 = \binom{52}{2} (2p)^{50} (-y)^2 = \frac{52 \cdot 51}{2} \cdot (2p)^{50} \cdot y^2 = 1326 \cdot (2p)^{50} \cdot y^2
\]

### Simplifying These Terms:

1. \((2p)^{52}\)
2. \(-52 \cdot (2p)^{51} \cdot y\)
3. \(1326 \cdot (2p)^{50} \cdot y^2\)

### Conclusion:

The first three terms of the binomial expansion of \((2p - y)^{52}\) are:

1. \((2p)^{52}\)
2. \(-52 \cdot (2p)^{51} \cdot y\)
3. \(1326 \cdot (2p)^{50} \cdot y^2\)

Would you like to see further details or have any questions?

Here are 5 related questions to explore further:
1. How would the expansion change if the exponent were different?
2. Can you calculate the next three terms of the expansion?
3. What are the binomial coefficients and how do they relate to Pascal's triangle?
4. How can binomial expansions be used to approximate values?
5. How does the sign of the terms change based on the binomial's second term?

**Tip:** In a binomial expansion, the signs of the terms alternate when the binomial has a subtraction sign (e.g., \((a - b)^n\)).

Explore the first 3 terms of the binomial expansion (2p - y)^52 using the binomial theorem. Understand how binomial coefficients and powers contribute to the expansion. Suitable for advanced high school students.

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