If `f(x)` is periodic with periot `'t'` such that `f(2x + 3) + f(2x+7) = 2`, then the coefficient of `m^-24` in the expansion of `(m+b/m^3)^(4t)` is

Options

A. `""^(16)C_(10) b^(6)`
B. `""^(16)C_(6) b^(10)`
C. `""^(16)C_(6) b^(4)`
D. `""^(16)C_(6)b^(6)`

Explanation

To solve the given problem step by step, we will start by analyzing the periodic function and then proceed to find the coefficient of \( m^{-24} \) in the expansion of \( \left(m + \frac{b}{m^3}\right)^{4t} \).

### Step 1: Understand the periodic function
We are given that:
\[ f(2x + 3) + f(2x + 7) = 2 \]
This indicates that the function \( f(x) \) is periodic.

### Step 2: Replace \( x \) with \( x + 2 \)
We replace \( x \) with \( x + 2 \) in the original equation:
\[ f(2(x + 2) + 3) + f(2(x + 2) + 7) = 2 \]
This simplifies to:
\[ f(2x + 4 + 3) + f(2x + 4 + 7) = 2 \]
or:
\[ f(2x + 7) + f(2x + 11) = 2 \]

### Step 3: Set up the equations
Now we have two equations:
1. \( f(2x + 3) + f(2x + 7) = 2 \) (Equation 1)
2. \( f(2x + 7) + f(2x + 11) = 2 \) (Equation 2)

### Step 4: Subtract the equations
Subtract Equation 1 from Equation 2:
\[ (f(2x + 7) + f(2x + 11)) - (f(2x + 3) + f(2x + 7)) = 0 \]
This simplifies to:
\[ f(2x + 11) - f(2x + 3) = 0 \]
Thus, we have:
\[ f(2x + 11) = f(2x + 3) \]

### Step 5: Determine the period
From the above equation, we can conclude that the function \( f(x) \) is periodic with a period of \( 8 \) (since \( 11 - 3 = 8 \)).

### Step 6: Find the coefficient of \( m^{-24} \)
Now we need to find the coefficient of \( m^{-24} \) in the expansion of:
\[ \left(m + \frac{b}{m^3}\right)^{4t} \]
Since \( t = 8 \), we have:
\[ \left(m + \frac{b}{m^3}\right)^{32} \]

### Step 7: Use the binomial theorem
Using the binomial theorem, we can expand:
\[ \left(m + \frac{b}{m^3}\right)^{32} = \sum_{r=0}^{32} \binom{32}{r} m^{32 - r} \left(\frac{b}{m^3}\right)^r \]
This simplifies to:
\[ = \sum_{r=0}^{32} \binom{32}{r} b^r m^{32 - r - 3r} = \sum_{r=0}^{32} \binom{32}{r} b^r m^{32 - 4r} \]

### Step 8: Set the exponent of \( m \)
We want the term where the exponent of \( m \) is \( -24 \):
\[ 32 - 4r = -24 \]
Solving for \( r \):
\[ 32 + 24 = 4r \]
\[ 56 = 4r \]
\[ r = 14 \]

### Step 9: Find the coefficient
Now, we need to find the coefficient of \( m^{-24} \):
\[ \text{Coefficient} = \binom{32}{14} b^{14} \]

### Final Answer
Thus, the coefficient of \( m^{-24} \) in the expansion is:
\[ \binom{32}{14} b^{14} \]