The largest coefficient in the expansion of `(1+x)^24` is

Options

A. `"^24C_24`
B. `"^24C_13`
C. `"^24C_12`
D. `"^24C_11`

Answer

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

### Step 1: Identify the General Term
The general term \(T_r\) in the expansion of \((1+x)^n\) is given by:
\[ T_r = \binom{n}{r} x^r \]
where \(n = 24\) and \(r\) varies from \(0\) to \(n\).

### Step 2: Write the General Term for the Given Expansion
For our case, the general term becomes:
\[ T_r = \binom{24}{r} x^r \]
The coefficient of this term is \(\binom{24}{r}\).

### Step 3: Determine the Range of \(r\)
The value of \(r\) can take any integer value from \(0\) to \(24\) (inclusive).

### Step 4: Analyze the Coefficients
The coefficients \(\binom{24}{r}\) increase as \(r\) increases from \(0\) to \(12\) and then decrease as \(r\) increases from \(12\) to \(24\). Therefore, the maximum coefficient occurs at \(r = 12\) or \(r = 11\) (since \(\binom{n}{r} = \binom{n}{n-r}\)).

### Step 5: Calculate the Largest Coefficient
To find the largest coefficient, we need to calculate:
\[ \binom{24}{12} \]
This can be computed using the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Thus,
\[ \binom{24}{12} = \frac{24!}{12! \cdot 12!} \]

### Step 6: Final Calculation
Calculating \(\binom{24}{12}\):
\[ \binom{24}{12} = \frac{24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \]
This will yield the largest coefficient in the expansion.

### Conclusion
Thus, the largest coefficient in the expansion of \((1+x)^{24}\) is \(\binom{24}{12}\).