Write the coefficient of the middle term in the expansion of `(1+x)^(2n)`.

Explanation

To find the coefficient of the middle term in the expansion of \( (1+x)^{2n} \), we can follow these steps:

### Step 1: Identify the total number of terms
In the expansion of \( (1+x)^{2n} \), the total number of terms is \( 2n + 1 \) because the expansion follows the binomial theorem.

**Hint:** The total number of terms in the expansion of \( (a+b)^n \) is given by \( n+1 \).

### Step 2: Determine the middle term
Since \( 2n \) is even, the middle term will be the \( (n+1) \)-th term. This is because the middle term in an expansion with an odd number of terms is located at the position \( \frac{(2n+1)}{2} \).

**Hint:** For an even power \( 2n \), the middle term is found at position \( n+1 \).

### Step 3: Write the formula for the \( k \)-th term
The \( k \)-th term in the expansion of \( (1+x)^{2n} \) is given by:
\[ T_k = \binom{2n}{k-1} x^{k-1} \]
where \( k \) is the term number.

**Hint:** The general term in the binomial expansion can be expressed using the binomial coefficient.

### Step 4: Substitute \( k = n+1 \)
To find the middle term, substitute \( k = n+1 \):
\[ T_{n+1} = \binom{2n}{n} x^n \]

**Hint:** The middle term corresponds to the term where \( k \) equals \( n+1 \).

### Step 5: Find the coefficient of the middle term
The coefficient of the middle term \( T_{n+1} \) is:
\[ \text{Coefficient} = \binom{2n}{n} \]

**Hint:** The coefficient of a term in the binomial expansion is given by the binomial coefficient.

### Final Answer
Thus, the coefficient of the middle term in the expansion of \( (1+x)^{2n} \) is:
\[ \binom{2n}{n} \]