In the expansion of `(x+y+z)^(20)`, which of the followign is False?

Options

A. Coefficient of `x^(7)y^(8)z^(7)` is zero
B. total number of distinct terms are `231`
C. every term of the form`(20!)/((20-r)!(r-k)!k!)`
`x^(20)y^(r-k)z^(k) (r,kepsilonW` and `r,kle20)`
D. total number of distinct terms are `221`

Explanation

To determine which statement is false in the expansion of \((x+y+z)^{20}\), we will analyze the properties of the multinomial expansion.

### Step-by-step Solution:

1. **Understanding the Expansion**:
The expansion of \((x+y+z)^{20}\) can be expressed using the multinomial theorem, which states that:
\[ (x+y+z)^n = \sum_{r_1 + r_2 + r_3 = n} \frac{n!}{r_1! r_2! r_3!} x^{r_1} y^{r_2} z^{r_3} \]
where \(n = 20\) in our case.

2. **Identifying Coefficients**:
Each term in the expansion corresponds to a combination of powers \(x^{r_1} y^{r_2} z^{r_3}\) such that \(r_1 + r_2 + r_3 = 20\). The coefficient of each term is given by:
\[ \frac{20!}{r_1! r_2! r_3!} \]

3. **Checking Specific Terms**:
Let's check the validity of the claims regarding specific terms:
- For the term \(x^7 y^8 z^5\), we have:
\[ r_1 = 7, \quad r_2 = 8, \quad r_3 = 5 \quad \Rightarrow \quad r_1 + r_2 + r_3 = 7 + 8 + 5 = 20 \]
This term is valid, and its coefficient is:
\[ \frac{20!}{7! 8! 5!} \]
- For the term \(x^7 y^8 z^7\), we have:
\[ r_1 = 7, \quad r_2 = 8, \quad r_3 = 7 \quad \Rightarrow \quad r_1 + r_2 + r_3 = 7 + 8 + 7 = 22 \]
This term is invalid since the sum exceeds 20. Therefore, the coefficient for this term is 0.

4. **Counting Distinct Terms**:
The total number of distinct terms in the expansion can be calculated using the formula for combinations with repetition:
\[ \text{Number of distinct terms} = \binom{n+k-1}{k-1} = \binom{20+3-1}{3-1} = \binom{22}{2} = \frac{22 \times 21}{2} = 231 \]

5. **Conclusion**:
Based on the analysis:
- The coefficient of \(x^7 y^8 z^7\) is indeed 0, making the statement about this term false.
- The total number of distinct terms is correctly calculated as 231.

### Final Answer:
The false statement is regarding the coefficient of \(x^7 y^8 z^7\) being non-zero.