To create a data set with a mean of 20, a median of 15, and a mode of 10, we can start by placing these specific values in the dataset. Let's denote the 7 numbers in the dataset as $a$ through $g$.
Given:
Mean = 20
Median = 15
Mode = 10
Step 1: Set up the data set
$a, b, c, 10, d, e, f$
Step 2: Find the mean
The mean of the data set can be calculated using the formula:
\text{Mean} = \frac{a + b + c + 10 + d + e + f}{7} = 20
a + b + c + d + e + f = 140 - 10
a + b + c + d + e + f = 130
Step 3: Find the median
Since we want the median to be 15 and the data set has 7 numbers, the 4th number in the ordered data set should be 15.
So, the ordered data set is:
$10, a, b, 15, c, d, e$
Step 4: Find the mode
Since the mode is 10, we already have one 10 in the data set.
Step 5: Solve for the missing values
From the mean calculation, we have:
a + b + c + d + e + f = 130
We can assign values to the variables to satisfy all the conditions:
$a = 5, b = 10, c = 15, d = 20, e = 25, f = 55$
So, the data set is:
$5, 10, 15, 10, 20, 25, 55$
\boxed{5, 10, 15, 10, 20, 25, 55} is the data set with the mean of 20, median of 15, and mode of 10.