If the orders of the two middle terms of the expansion of (x + y)n are 7 and 8, then n equals:

The problem states that the orders of the two middle terms of the expansion of \((x + y)^n\) are 7 and 8. We need to determine the value of \( n \).

### Step 1: Understanding the middle terms
For the binomial expansion \((x + y)^n\), the general term is:
\[
T_{r+1} = \binom{n}{r} x^{n-r}y^r
\]
where \( r \) is the order of the term.

### Step 2: Finding the middle terms
If \( n \) is even, the two middle terms are the terms at positions \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \).
If \( n \) is odd, the middle term is the one at position \( \frac{n+1}{2} \).

Given that the orders of the two middle terms are 7 and 8, it means that the two middle terms are consecutive, indicating that \( n \) is even. Let's set up the equation for the two middle terms:
\[
\frac{n}{2} = 7 \quad \text{and} \quad \frac{n}{2} + 1 = 8
\]

### Step 3: Solving for \( n \)
Solving \(\frac{n}{2} = 7\):
\[
n = 2 \times 7 = 14
\]

Thus, the value of \( n \) is:
\[
n = 14
\]

### Conclusion
The correct answer is \( n = 14 \).

Would you like more details or have any other questions?

Here are some related questions to deepen your understanding:

1. What are the conditions for a binomial expansion to have two middle terms?
2. How do you find the general term in the expansion of \((x + y)^n\)?
3. What happens to the number of middle terms when \(n\) is odd in a binomial expansion?
4. How do you calculate the coefficients of specific terms in a binomial expansion?
5. What is the significance of symmetry in the binomial coefficients?

**Tip:** In a binomial expansion, the sum of the exponents in each term always equals the power \(n\). This can help verify if terms are correctly identified.

If the orders of the two middle terms of the expansion of (x + y)^n are 7 and 8, then n equals:

Learn how to determine the value of n in the binomial expansion (x + y)^n when the orders of the two middle terms are 7 and 8. This problem involves binomial expansion concepts and the binomial theorem. Suitable for students in Grades 10-12.

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