If the sum of the coefficients in the expansion of
`(1 + 2x)^(n)` is 6561 , then the greatest coefficients in the expansion, is

Options

A. 896
B. 3594
C. 1792
D. none of these

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understanding the Sum of Coefficients
The sum of the coefficients in the expansion of \( (1 + 2x)^n \) can be found by substituting \( x = 1 \):
\[ (1 + 2 \cdot 1)^n = (1 + 2)^n = 3^n \]

### Step 2: Setting Up the Equation
We know from the problem that the sum of the coefficients is equal to 6561. Therefore, we can set up the equation:
\[ 3^n = 6561 \]

### Step 3: Solving for \( n \)
Next, we need to express 6561 as a power of 3. We can find that:
\[ 6561 = 3^8 \]
Thus, we have:
\[ 3^n = 3^8 \implies n = 8 \]

### Step 4: Finding the Greatest Coefficient
To find the greatest coefficient in the expansion of \( (1 + 2x)^n \), we will use the formula for the \( r \)-th term in the binomial expansion:
\[ T_r = \binom{n}{r} (1)^{n-r} (2x)^r = \binom{n}{r} 2^r x^r \]
We need to find the value of \( r \) that maximizes \( T_r \).

### Step 5: Using the Ratio of Consecutive Terms
To find the maximum term, we can use the ratio of consecutive terms:
\[ \frac{T_{r+1}}{T_r} = \frac{\binom{n}{r+1} 2^{r+1}}{\binom{n}{r} 2^r} = \frac{n - r}{r + 1} \cdot 2 \]
Setting this ratio greater than 1 for maximization:
\[ \frac{n - r}{r + 1} \cdot 2 > 1 \]
Substituting \( n = 8 \):
\[ \frac{8 - r}{r + 1} \cdot 2 > 1 \]

### Step 6: Solving the Inequality
Now, we solve the inequality:
\[ 2(8 - r) > r + 1 \implies 16 - 2r > r + 1 \implies 16 - 1 > 3r \implies 15 > 3r \implies r < 5 \]
Thus, \( r \) can be 5 or 6.

### Step 7: Calculating the Greatest Coefficients
Now we calculate the coefficients for \( r = 5 \) and \( r = 6 \):
1. For \( r = 5 \):
\[ T_5 = \binom{8}{5} 2^5 = \binom{8}{3} 2^5 = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} \cdot 32 = 56 \cdot 32 = 1792 \]

2. For \( r = 6 \):
\[ T_6 = \binom{8}{6} 2^6 = \binom{8}{2} 2^6 = \frac{8 \cdot 7}{2 \cdot 1} \cdot 64 = 28 \cdot 64 = 1792 \]

### Conclusion
Both \( T_5 \) and \( T_6 \) yield the same value, which is 1792. Therefore, the greatest coefficient in the expansion is:
\[ \boxed{1792} \]