The number of distinct terms in the expansion of is `(x^(3)+(1)/(x^(3))+1)^(200)` is

Options

A. `201`
B. `400`
C. `401`
D. `500`

Explanation

To find the number of distinct terms in the expansion of \((x^3 + \frac{1}{x^3} + 1)^{200}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (x^3 + \frac{1}{x^3} + 1)^{200} \]
We can factor out \(\frac{1}{x^3}\) from the terms:
\[ = \left(\frac{1}{x^3}(x^6 + 1 + x^3)\right)^{200} \]
This simplifies to:
\[ = \frac{1}{x^{600}}(x^6 + x^3 + 1)^{200} \]

### Step 2: Substitute for simplification
Let \(t = x^3\). Then, we can rewrite the expression as:
\[ = \frac{1}{x^{600}}(t^2 + t + 1)^{200} \]

### Step 3: Analyze the polynomial
Now we need to find the number of distinct terms in the expansion of \((t^2 + t + 1)^{200}\). The polynomial \(t^2 + t + 1\) is a quadratic polynomial.

### Step 4: Determine the maximum degree
The highest degree of the polynomial \(t^2 + t + 1\) is 2. When raised to the power of 200, the maximum degree of the expansion will be:
\[ 2 \times 200 = 400 \]

### Step 5: Count the distinct terms
The number of distinct terms in the expansion of a polynomial of degree \(n\) is given by \(n + 1\). Therefore, for our polynomial of degree 400:
\[ \text{Number of distinct terms} = 400 + 1 = 401 \]

### Step 6: Conclusion
Thus, the number of distinct terms in the expansion of \((x^3 + \frac{1}{x^3} + 1)^{200}\) is:
\[ \boxed{401} \]

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