Find the Impossible Average: Analyzing Numbers 8, 7, 5, and 10

To solve this problem, we'll follow these steps:

• Step 1: Calculate the sum of the numbers.
• Step 2: Find the average by dividing this sum by the number of numbers.
• Step 3: Compare this calculated average to the given options to identify which cannot be the average.

Now, let's work through each step:
Step 1: Calculate the sum of the numbers: $8 + 7 + 5 + 10 = 30$.
Step 2: Find the average by dividing the sum by the total number of numbers: $\frac{30}{4} = 7.5$.
Step 3: Compare the calculated average to the options provided:

• Option 1: $6$ - not equal to $7.5$.
• Option 2: $8$ - not equal to $7.5$, but close enough to an integer comparison standard.
• Option 3: $12$ - clearly not possible since $7.5 < 12$.
• Option 4: $9$ - not equal to $7.5$.

Of all these options, the choice that cannot possibly be the average based on the numbers given is $12$. Therefore, the answer is $12$.