The sum of the coefficients in the expansion of `(x^(2)- (1)/( 3))^(199) xx ( x^(3) + ( 1)/( 2))^(200)` is

Options

A. `1//3`
B. `-1//3`
C. `2//3`
D. `3//2`

Answer

To find the sum of the coefficients in the expansion of \((x^2 - \frac{1}{3})^{199} \cdot (x^3 + \frac{1}{2})^{200}\), we can follow these steps:

### Step 1: Substitute \(x = 1\)
To find the sum of the coefficients, we substitute \(x = 1\) into the expression.

\[ (1^2 - \frac{1}{3})^{199} \cdot (1^3 + \frac{1}{2})^{200} \]

### Step 2: Simplify the expressions
Now, simplify each part:

1. For \((1^2 - \frac{1}{3})^{199}\):
\[ 1 - \frac{1}{3} = \frac{2}{3} \]
So,
\[ (1 - \frac{1}{3})^{199} = \left(\frac{2}{3}\right)^{199} \]

2. For \((1^3 + \frac{1}{2})^{200}\):
\[ 1 + \frac{1}{2} = \frac{3}{2} \]
So,
\[ (1 + \frac{1}{2})^{200} = \left(\frac{3}{2}\right)^{200} \]

### Step 3: Combine the results
Now, we combine the two results from above:

\[ \left(\frac{2}{3}\right)^{199} \cdot \left(\frac{3}{2}\right)^{200} \]

### Step 4: Simplify the combined expression
We can rewrite this as:

\[ \frac{2^{199}}{3^{199}} \cdot \frac{3^{200}}{2^{200}} = \frac{2^{199} \cdot 3^{200}}{3^{199} \cdot 2^{200}} \]

### Step 5: Simplify further
Now, simplify the expression:

\[ = \frac{3^{200}}{3^{199}} \cdot \frac{2^{199}}{2^{200}} = \frac{3^{200}}{3^{199}} \cdot \frac{1}{2} = 3 \cdot \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, the sum of the coefficients in the expansion is:

\[ \frac{3}{2} \]

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