Find the largest binomial coefficients in the
expansion of `(1 + x)^24`

Answer

To find the largest binomial coefficient in the expansion of \((1 + x)^{24}\), we can follow these steps:

### Step 1: Identify \(n\)
The given expression is \((1 + x)^{24}\). Here, \(n = 24\).

### Step 2: Determine if \(n\) is even or odd
Since \(n = 24\) is an even number, we can use the formula for the largest binomial coefficient.

### Step 3: Use the formula for the largest binomial coefficient
For an even \(n\), the largest binomial coefficient is given by:
\[ \binom{n}{n/2} \]
In our case, since \(n = 24\):
\[ \text{Largest binomial coefficient} = \binom{24}{24/2} = \binom{24}{12} \]

### Step 4: Calculate \(\binom{24}{12}\)
To find \(\binom{24}{12}\), we use the formula for binomial coefficients:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Substituting \(n = 24\) and \(r = 12\):
\[ \binom{24}{12} = \frac{24!}{12! \cdot 12!} \]

### Step 5: Conclusion
The largest binomial coefficient in the expansion of \((1 + x)^{24}\) is \(\binom{24}{12}\).

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