Find the coefficient of one middle term in the expansio of `(1+x)^(2n)`.

Answer

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 given by \(2n + 1\) because the general formula for the number of terms in the expansion of \((a + b)^m\) is \(m + 1\).

**Hint:** Remember that for any binomial expansion \((a + b)^m\), the number of terms is \(m + 1\).

### Step 2: Determine the middle term
Since \(2n\) is an even number, 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 \(\frac{(2n + 1)}{2} = n + 1\).

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

### Step 3: Write the general term
The general term (T_r) in the expansion of \((1+x)^{2n}\) is given by:
\[ T_{r+1} = \binom{2n}{r} x^r \]
where \(r\) is the term number starting from 0.

**Hint:** The general term formula for binomial expansion is \(\binom{n}{r} a^{n-r} b^r\).

### Step 4: Find the middle term
For the middle term, we substitute \(r = n\) (since \(n + 1\) corresponds to \(r = n\)):
\[ T_{n+1} = \binom{2n}{n} x^n \]

**Hint:** Substitute \(r = n\) to find the middle term.

### Step 5: Identify the coefficient of the middle term
The coefficient of the middle term is simply \(\binom{2n}{n}\).

**Hint:** The coefficient of any term in a 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:
\[ \binom{2n}{n} \]