If the sum of the coefficients in the expansion of `(x +y)^(n)` is 2048, then the greatest coefficient in the expansion is:

Options

A. `""^(10)C_(6)`
B. `""^(11)C_(6)`
C. `""^(11)C_(7)`
D. `""^(12)C_(6)`

Answer

To solve the problem step by step, we need to find the greatest coefficient in the expansion of \((x + y)^n\) given that the sum of the coefficients is 2048.

### Step 1: Understand the sum of coefficients
The sum of the coefficients in the expansion of \((x + y)^n\) can be found by substituting \(x = 1\) and \(y = 1\):
\[ (1 + 1)^n = 2^n \]
Thus, the sum of the coefficients is \(2^n\).

### Step 2: Set up the equation
According to the problem, the sum of the coefficients is 2048. Therefore, we have:
\[ 2^n = 2048 \]

### Step 3: Solve for \(n\)
To find \(n\), we need to express 2048 as a power of 2. We can calculate:
\[ 2048 = 2^{11} \]
Thus, we can equate:
\[ n = 11 \]

### Step 4: Identify the greatest coefficient
In the expansion of \((x + y)^{11}\), the coefficients are given by the binomial coefficients \(\binom{11}{k}\) for \(k = 0, 1, 2, \ldots, 11\). The greatest coefficient occurs at the middle of the expansion.

### Step 5: Determine the position of the greatest coefficient
Since \(n = 11\) is odd, the greatest coefficients will be at the terms:
- \(\binom{11}{5}\) (6th term)
- \(\binom{11}{6}\) (7th term)

### Step 6: Calculate the greatest coefficients
Now we can calculate:
\[ \binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11!}{5!6!} \]
Calculating this gives:
\[ \binom{11}{5} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462 \]

Similarly,
\[ \binom{11}{6} = \frac{11!}{6!(11-6)!} = \frac{11!}{6!5!} = 462 \]

### Conclusion
Thus, the greatest coefficient in the expansion of \((x + y)^{11}\) is:
\[ \text{Greatest Coefficient} = 462 \]