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 need to analyze the coefficients and the distinct terms in the expansion. Let's go through the steps systematically.

### Step 1: Understanding the Expansion
The expansion of \((x+y+z)^{20}\) can be expressed using the multinomial theorem. The general term in the expansion is given by:

\[ \frac{20!}{r_1! r_2! r_3!} x^{r_1} y^{r_2} z^{r_3} \]

where \(r_1 + r_2 + r_3 = 20\).

### Step 2: Analyzing the Coefficient of a Specific Term
Let's check the coefficient of the term \(x^7 y^8 z^5\):

1. Calculate the sum of the powers:
\[ r_1 + r_2 + r_3 = 7 + 8 + 5 = 20 \]

2. Since the sum equals 20, we can find the coefficient:
\[ \text{Coefficient} = \frac{20!}{7!8!5!} \]

3. This coefficient is not zero, so the statement regarding this term is likely true.

### Step 3: Counting Distinct Terms
The number of distinct terms in the expansion can be calculated using the formula for combinations:

\[ \text{Number of distinct terms} = \binom{n+k-1}{k-1} \]

where \(n\) is the power (20) and \(k\) is the number of variables (3). Thus, we have:

\[ \text{Number of distinct terms} = \binom{20+3-1}{3-1} = \binom{22}{2} \]

Calculating this gives:

\[ \binom{22}{2} = \frac{22 \times 21}{2 \times 1} = 231 \]

### Step 4: Evaluating Each Statement
Now, we need to evaluate the options given in the question:

1. **Coefficient of \(x^7 y^8 z^5\)**: As calculated, this coefficient is non-zero.
2. **Total number of distinct terms**: We found this to be 231.
3. **Form of the term**: The general form of the term must be \(x^{20-r} y^{r-k} z^k\). If any option states otherwise, it could be false.

### Final Step: Identifying the False Statement
From the analysis:

- The coefficient of \(x^7 y^8 z^5\) is valid.
- The total number of distinct terms is correctly calculated as 231.
- If any statement incorrectly represents the form of the term, that would be the false statement.

### Conclusion
The false statement is the one that misrepresents the form of the term in the expansion. Based on the analysis, we conclude that the false statement is related to the incorrect expression of the term's powers or coefficients.