The number of terms in the expansion of `(1+2x +x^2)^20` is 40

Answer

To determine whether the statement "The number of terms in the expansion of \( (1 + 2x + x^2)^{20} \) is 40" is true or false, we will analyze the expression step by step.

### Step-by-Step Solution:

1. **Identify the expression**: We start with the expression \( (1 + 2x + x^2)^{20} \).

2. **Recognize the form**: This expression can be viewed as a trinomial expansion. The general form of a trinomial expansion \( (a + b + c)^n \) can be analyzed using the multinomial theorem.

3. **Determine the number of distinct terms**: The number of distinct terms in the expansion of \( (a + b + c)^n \) is given by the formula:
\[ \text{Number of terms} = \frac{(n + k - 1)!}{n!(k - 1)!} \]
where \( n \) is the exponent and \( k \) is the number of different terms in the expression.

4. **Apply the formula**: In our case, \( n = 20 \) and \( k = 3 \) (since we have the terms \( 1, 2x, \) and \( x^2 \)).
\[ \text{Number of terms} = \frac{(20 + 3 - 1)!}{20!(3 - 1)!} = \frac{22!}{20! \cdot 2!} \]

5. **Calculate the factorials**:
\[ = \frac{22 \times 21}{2 \times 1} = \frac{462}{2} = 231 \]

6. **Conclusion**: The number of distinct terms in the expansion of \( (1 + 2x + x^2)^{20} \) is 231, not 40. Therefore, the statement is false.

### Final Answer:
The statement "The number of terms in the expansion of \( (1 + 2x + x^2)^{20} \) is 40" is **false**.