The given expression is $(\frac{a}{b} + \frac{b}{a})^6$.

To find the middle term in the binomial expansion of $(x + y)^n$, the general formula for the $r$-th term is:

$T_{r+1} = \binom{n}{r} \cdot x^{n-r} \cdot y^r$

Here, $n = 6$, $x = \frac{a}{b}$, and $y = \frac{b}{a}$.

Since $n = 6$, the total number of terms is $n + 1 = 7$. The middle term is the 4th term, which corresponds to $r = 3$.

Substituting into the formula for the general term:

$T_4 = \binom{6}{3} \cdot \left( \frac{a}{b} \right)^{6-3} \cdot \left( \frac{b}{a} \right)^3$

Now, simplify the powers:

$T_4 = \binom{6}{3} \cdot \left( \frac{a}{b} \right)^3 \cdot \left( \frac{b}{a} \right)^3$

Since $\left( \frac{a}{b} \right)^3 \cdot \left( \frac{b}{a} \right)^3 = 1$, the term simplifies to:

$T_4 = \binom{6}{3} \cdot 1$

Now, calculate $\binom{6}{3}$:

$\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$

Thus, the middle term is:

$T_4 = 20$

Final Answer:

The middle term in the expansion of $(\frac{a}{b} + \frac{b}{a})^6$ is 20.

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Do you want more details or have any questions?

Here are some related questions to explore further:

1. What is the general term for the binomial expansion of $(a+b)^n$?
2. How can we find the sum of all the coefficients in a binomial expansion?
3. What would be the middle term in $(x + y)^{10}$?
4. How does Pascal’s Triangle relate to binomial expansions?
5. How do you find the coefficient of a specific term in binomial expansions?

Tip: Remember, the middle term in a binomial expansion occurs at $\frac{n+1}{2}$-th position when $n$ is even.