The greatest coefficient in `((x^(3//2)y)/(2) + (2)/(xy^(3//2)))^12` is

Options

A. `12(2^11)`
B. `12(2^10)`
C. `12(2^22)`
D. `33(2^9)`

Answer

To find the greatest coefficient in the expression \(\left(\frac{x^{3/2}y}{2} + \frac{2}{xy^{3/2}}\right)^{12}\), we will use the Binomial Theorem. Here are the steps to solve the problem:

### Step 1: Identify the terms in the binomial expansion
The expression can be rewritten as:
\[ \left(a + b\right)^{n} \]
where \(a = \frac{x^{3/2}y}{2}\) and \(b = \frac{2}{xy^{3/2}}\), and \(n = 12\).

### Step 2: Apply the Binomial Theorem
According to the Binomial Theorem, the expansion of \((a + b)^n\) is given by:
\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \]

### Step 3: Find the general term
The general term \(T_k\) in the expansion can be expressed as:
\[ T_k = \binom{n}{k} a^{n-k} b^{k} \]
Substituting the values of \(a\) and \(b\):
\[ T_k = \binom{12}{k} \left(\frac{x^{3/2}y}{2}\right)^{12-k} \left(\frac{2}{xy^{3/2}}\right)^{k} \]

### Step 4: Simplify the general term
Now simplify \(T_k\):
\[ T_k = \binom{12}{k} \left(\frac{1}{2}\right)^{12-k} \left(\frac{2^k}{x^k y^{3k/2}}\right) \left(x^{(3/2)(12-k)} y^{(12-k)}\right) \]
Combining the terms:
\[ T_k = \binom{12}{k} \cdot \frac{2^k}{2^{12-k}} \cdot x^{(3/2)(12-k) - k} \cdot y^{(12-k) - \frac{3k}{2}} \]
This simplifies to:
\[ T_k = \binom{12}{k} \cdot \frac{2^k}{2^{12}} \cdot x^{18 - \frac{5k}{2}} \cdot y^{12 - \frac{5k}{2}} \]

### Step 5: Determine the exponent conditions
To find the greatest coefficient, we need to find the value of \(k\) that maximizes the coefficient. The coefficient of \(T_k\) is:
\[ \text{Coefficient} = \binom{12}{k} \cdot \frac{2^k}{2^{12}} = \binom{12}{k} \cdot \frac{1}{2^{12-k}} \]

### Step 6: Find the maximum coefficient
The maximum coefficient occurs at \(k = \frac{n}{2} = \frac{12}{2} = 6\). Thus, we need to calculate:
\[ \binom{12}{6} \]

### Step 7: Calculate \(\binom{12}{6}\)
Using the formula for combinations:
\[ \binom{12}{6} = \frac{12!}{6!6!} = \frac{479001600}{720 \cdot 720} = 924 \]

### Final Answer
The greatest coefficient in the expansion is:
\[ \boxed{924} \]