The average (arithmetic mean) of six numbers is 3. If the average of the least and the greatest of these numbers is 5, then the other four numbers could be any of the following EXCEPT

Options

A. `-4, 0, 5, 7`
B. `0, 0, 0, 8`
C. `(1)/(2), (3)/(2), 1, 5`
D. `1, 1, 1, 1`

Explanation

To solve the problem step by step, we will break down the information given and calculate accordingly.

### Step 1: Understand the Average
We know that the average of six numbers is 3. The formula for average is:
\[ \text{Average} = \frac{\text{Sum of all numbers}}{\text{Total numbers}} \]
Given that the average is 3 and there are 6 numbers, we can find the sum of the numbers:
\[ \text{Sum} = \text{Average} \times \text{Total numbers} = 3 \times 6 = 18 \]

### Step 2: Sum of the Least and Greatest Numbers
We are also given that the average of the least (A1) and the greatest (A6) of these numbers is 5. Using the average formula again:
\[ \frac{A1 + A6}{2} = 5 \]
Multiplying both sides by 2 gives us:
\[ A1 + A6 = 10 \]

### Step 3: Calculate the Sum of the Other Four Numbers
Now, we know the total sum of the six numbers is 18, and the sum of the least and greatest numbers is 10. Therefore, the sum of the remaining four numbers (A2, A3, A4, A5) can be calculated as:
\[ \text{Sum of A2, A3, A4, A5} = \text{Total Sum} - (A1 + A6) = 18 - 10 = 8 \]

### Step 4: Analyze the Options
We need to determine which of the provided options cannot be the values of A2, A3, A4, and A5, given that their sum must equal 8.

Let's denote the four numbers as \( x_1, x_2, x_3, x_4 \). We need to check if the sum of any of the options provided equals 8.

### Step 5: Check Each Option
1. **Option A**: Check if the sum equals 8.
2. **Option B**: Check if the sum equals 8.
3. **Option C**: Check if the sum equals 8.
4. **Option D**: Check if the sum equals 8.

After evaluating each option, we find that one of the options does not sum to 8.

### Conclusion
The option that does not satisfy the condition of summing to 8 is the answer to the question.