The number of distinct terms in the expansion of is `(x^(3)+(1)/(x^(3))+1)^(200)` is (a) 201 (b) 400 (c) 401 (d) 500

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: Identify the expression
We start with the expression:
\[ (x^3 + \frac{1}{x^3} + 1)^{200} \]

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

### Step 3: Analyze the binomial expansion
The expression \((t + \frac{1}{t} + 1)^{200}\) can be expanded using the multinomial theorem. The general term in the expansion can be expressed as:
\[ \frac{200!}{a!b!c!} t^a \left(\frac{1}{t}\right)^b 1^c \]
where \( a + b + c = 200 \).

### Step 4: Simplify the general term
The general term simplifies to:
\[ \frac{200!}{a!b!c!} t^{a-b} \]
This means that the exponent of \( t \) (which is \( x^3 \)) will be \( a - b \).

### Step 5: Determine the range of \( a - b \)
The values of \( a \) and \( b \) can range from \( 0 \) to \( 200 \). Therefore:
- The minimum value of \( a - b \) occurs when \( a = 0 \) and \( b = 200 \), giving \( a - b = 0 - 200 = -200 \).
- The maximum value of \( a - b \) occurs when \( a = 200 \) and \( b = 0 \), giving \( a - b = 200 - 0 = 200 \).

### Step 6: Calculate the distinct values of \( a - b \)
The possible values of \( a - b \) range from \( -200 \) to \( 200 \). This gives us:
\[ 200 - (-200) + 1 = 200 + 200 + 1 = 401 \]

### Conclusion
Thus, the number of distinct terms in the expansion of \((x^3 + \frac{1}{x^3} + 1)^{200}\) is \( 401 \).

### Final Answer
The answer is \( \boxed{401} \).