Solution:

The given equation is xyz = 10.

To find the number of solutions in positive integers x, y, z, we need to consider the factors of 10.

The prime factorization of 10 is 2 * 5.

This means that 10 can be written as the product of 2 and 5.

So, the possible values of x, y, and z are as follows:

- If x = 1, y = 1, and z = 10, then xyz = 1 * 1 * 10 = 10.
- If x = 1, y = 2, and z = 5, then xyz = 1 * 2 * 5 = 10.
- If x = 1, y = 5, and z = 2, then xyz = 1 * 5 * 2 = 10.
- If x = 1, y = 10, and z = 1, then xyz = 1 * 10 * 1 = 10.
- If x = 2, y = 1, and z = 5, then xyz = 2 * 1 * 5 = 10.
- If x = 2, y = 5, and z = 1, then xyz = 2 * 5 * 1 = 10.
- If x = 5, y = 1, and z = 2, then xyz = 5 * 1 * 2 = 10.
- If x = 5, y = 2, and z = 1, then xyz = 5 * 2 * 1 = 10.
- If x = 10, y = 1, and z = 1, then xyz = 10 * 1 * 1 = 10.

From the above values, we can observe that there are 9 unique solutions for x, y, and z.

However, we need to consider the order of the variables.

For example, (x = 1, y = 2, z = 5) is the same solution as (x = 2, y = 1, z = 5).

To find the total number of solutions, we need to find the number of unique permutations of x, y, and z.

Since there are 3 variables, the total number of permutations is 3!

3! = 3 * 2 * 1 = 6.

Therefore, the number of solutions is 9 * 6 = 54.

But we also need to consider the case when all three variables are equal to 10.

In that case, the equation becomes 10 * 10 * 10 = 1000.

So, there is one more solution.

Therefore, the total number of solutions is 54 + 1 = 55.

Hence, the correct answer is option 'B' - 55.