Sum of coefficients of expansion of B is 6561 . The difference of
the coefficient of third to the second term in the expansion of A is equal to 117 .
If `n^(m)` is divided by 7 , the remainder is

Options

A. 1
B. 2
C. 3
D. 5

Explanation

To solve the problem step by step, we will break down the information given and apply the Binomial Theorem.

### Step 1: Understanding the Sum of Coefficients
The sum of the coefficients of the binomial expansion \( B = \left( \frac{5}{2} + \frac{x}{2} \right)^n \) can be found by substituting \( x = 1 \):
\[ B(1) = \left( \frac{5}{2} + \frac{1}{2} \right)^n = \left( \frac{6}{2} \right)^n = 3^n \]
Given that this sum equals 6561, we can set up the equation:
\[ 3^n = 6561 \]

### Step 2: Solving for \( n \)
To find \( n \), we need to express 6561 as a power of 3. We can calculate:
\[ 6561 = 3^8 \]
Thus, we have:
\[ n = 8 \]

### Step 3: Coefficients of the Terms in Expansion of \( A \)
Next, we analyze the expansion of \( A = (1 + 3x)^m \). The coefficients of the terms can be expressed using the binomial coefficients:
- The coefficient of the second term \( T_2 \) is given by \( \binom{m}{1} \cdot 3^1 = 3m \).
- The coefficient of the third term \( T_3 \) is given by \( \binom{m}{2} \cdot 3^2 = \frac{m(m-1)}{2} \cdot 9 \).

### Step 4: Setting Up the Equation
According to the problem, the difference between the coefficients of the third and second terms is 117:
\[ T_3 - T_2 = 117 \]
Substituting the expressions for \( T_3 \) and \( T_2 \):
\[ \frac{m(m-1)}{2} \cdot 9 - 3m = 117 \]

### Step 5: Simplifying the Equation
Multiplying through by 2 to eliminate the fraction:
\[ 9m(m-1) - 6m = 234 \]
This simplifies to:
\[ 9m^2 - 15m - 234 = 0 \]

### Step 6: Solving the Quadratic Equation
We can use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 9, b = -15, c = -234 \):
\[ m = \frac{15 \pm \sqrt{(-15)^2 - 4 \cdot 9 \cdot (-234)}}{2 \cdot 9} \]
Calculating the discriminant:
\[ = 225 + 8412 = 8637 \]
Now, we compute \( m \):
\[ m = \frac{15 \pm \sqrt{8637}}{18} \]
Calculating \( \sqrt{8637} \approx 93 \):
\[ m = \frac{15 \pm 93}{18} \]
This gives two possible values:
1. \( m = \frac{108}{18} = 6 \)
2. \( m = \frac{-78}{18} \) (not valid as \( m \) must be positive)

Thus, \( m = 6 \).

### Step 7: Finding \( n^m \mod 7 \)
Now we need to compute \( n^m \mod 7 \):
\[ n^m = 8^6 \]
Calculating \( 8 \mod 7 \):
\[ 8 \equiv 1 \mod 7 \]
Thus:
\[ 8^6 \equiv 1^6 \equiv 1 \mod 7 \]

### Final Answer
The remainder when \( n^m \) is divided by 7 is:
\[ \boxed{1} \]