We are given the following information:

• The universal set $S$ has 215 elements.
• Set $A$ has 70 elements: $|A| = 70$.
• Set $B$ has 99 elements: $|B| = 99$.
• The number of elements in either $A$ or $B$ is 168: $|A \cup B| = 168$.

We are asked to find the number of elements in $A$ but not in $B$, denoted by $|A - B|$.

Key Formula:

We can use the principle of inclusion and exclusion to solve this. The formula for the union of two sets is: $|A \cup B| = |A| + |B| - |A \cap B|$ Substitute the known values into the formula: $168 = 70 + 99 - |A \cap B|$ Simplifying this: $168 = 169 - |A \cap B|$ $|A \cap B| = 169 - 168 = 1$

So, the number of elements in both $A$ and $B$ is $|A \cap B| = 1$.

Now, to find $|A - B|$:

The number of elements in $A$ but not in $B$ is: $|A - B| = |A| - |A \cap B|$ Substitute the known values: $|A - B| = 70 - 1 = 69$

Thus, the number of elements in $A$ but not in $B$ is $\boxed{69}$.

Would you like more details or have any questions? Here are some related questions you can explore:

1. How many elements are in $B - A$?
2. How many elements are in $A \cap B$?
3. What is the number of elements in the complement of $A \cup B$?
4. If $|S|$ were different, how would that affect $A$ and $B$?
5. Can you generalize this problem for any two sets $A$ and $B$?

Tip: Always verify if you need the intersection or complement of sets when dealing with set operations!