Solution:

To find the total number of positive integer solutions to the equation

(x1 x2 x3)(y1 y2 y3 y4) = 15,

we can start by finding all the possible factorizations of 15 into two positive integers.

Step 1: Factorization of 15

The prime factorization of 15 is:

15 = 3 * 5

So, the possible factorizations of 15 into two positive integers are:

15 * 1
5 * 3

Step 2: Finding the number of positive integer solutions

Now, let's consider each factorization separately and find the number of positive integer solutions for each case.

Case 1: (x1 x2 x3) = 15 and (y1 y2 y3 y4) = 1

In this case, we have one possible solution:

x1 = 15, x2 = 1, x3 = 1
y1 = 1, y2 = 1, y3 = 1, y4 = 1

So, there is 1 positive integer solution for this case.

Case 2: (x1 x2 x3) = 5 and (y1 y2 y3 y4) = 3

In this case, we have three possible solutions:

x1 = 5, x2 = 1, x3 = 1
y1 = 3, y2 = 1, y3 = 1, y4 = 1

x1 = 1, x2 = 5, x3 = 1
y1 = 3, y2 = 1, y3 = 1, y4 = 1

x1 = 1, x2 = 1, x3 = 5
y1 = 3, y2 = 1, y3 = 1, y4 = 1

So, there are 3 positive integer solutions for this case.

Case 3: (x1 x2 x3) = 3 and (y1 y2 y3 y4) = 5

In this case, we have three possible solutions:

x1 = 3, x2 = 1, x3 = 1
y1 = 5, y2 = 1, y3 = 1, y4 = 1

x1 = 1, x2 = 3, x3 = 1
y1 = 5, y2 = 1, y3 = 1, y4 = 1

x1 = 1, x2 = 1, x3 = 3
y1 = 5, y2 = 1, y3 = 1, y4 = 1

So, there are 3 positive integer solutions for this case.

Step 3: Total number of positive integer solutions

To find the total number of positive integer solutions, we add up the number of solutions from each case:

1 + 3 + 3 = 7

Therefore, there are a total of 7 positive integer solutions to the equation (x1 x2 x3)(y1 y2 y3 y4) = 15.