Introduction:
In this question, we are given a population consisting of elements that can be classified according to the presence or absence of three attributes, namely A, B, and C. We need to determine the number of smallest ultimate classes into which the population can be divided based on these attributes.

Understanding the problem:
To solve this problem, we need to understand the concept of ultimate classes. Ultimate classes are the smallest possible groups or divisions that can be formed based on the given attributes. Each ultimate class represents a unique combination of attribute presence or absence.

Solution:
To determine the number of smallest ultimate classes, we can use the concept of Cartesian product. The Cartesian product of three sets, each representing the presence or absence of attributes A, B, and C, will give us all possible combinations of these attributes.

Step 1: Create the sets:
Let's create three sets - A, B, and C, representing the presence or absence of attributes A, B, and C, respectively.

A = {1, 0} (1 represents attribute A present, 0 represents attribute A absent)
B = {1, 0} (1 represents attribute B present, 0 represents attribute B absent)
C = {1, 0} (1 represents attribute C present, 0 represents attribute C absent)

Step 2: Find the Cartesian product:
Now, find the Cartesian product of these sets. It can be computed by taking all possible combinations of one element from each set.

A × B × C = {(1, 1, 1), (1, 1, 0), (1, 0, 1), (1, 0, 0), (0, 1, 1), (0, 1, 0), (0, 0, 1), (0, 0, 0)}

Step 3: Determine the number of ultimate classes:
The Cartesian product gives us all possible combinations of the attributes. Each combination represents an ultimate class. Therefore, the number of smallest ultimate classes is equal to the cardinality of the Cartesian product.

In this case, the Cartesian product has 8 elements. Hence, the number of smallest ultimate classes into which the population can be divided is 8.

Conclusion:
The population can be divided into 8 smallest ultimate classes based on the presence or absence of attributes A, B, and C. The ultimate classes represent all possible combinations of these attributes. By finding the Cartesian product of the sets representing these attributes, we can determine the number of ultimate classes.