Understanding the Problem
We have two groups here:
1. Drama Club: Total of 75 members.
2. Students Union: Total of 90 students.

There's an overlap between these two groups: some drama club members are also in the students union. The problem gives us a specific condition about this overlap:
- At most 70 of the drama club members are in the students union.

This means the number of drama club members who are also in the students union is $\leq 70$.
We're asked: What could be the number of members of the students union who are not in the drama club?

In other words, how many students are in the students union but not in the drama club? And which of the given options (A to E) are possible values for this number?

Defining Variables
Let's define some variables to make this clearer:
- Let $D$ be the set of drama club members. $|D|=75$.
- Let $S$ be the set of students union members. $|S|=90$.
- Let $D \cap S$ be the set of students who are in both the drama club and the students union.

Given: $\(|D \cap S| \leq 70\)$.
We're interested in the number of students who are in the students union but not in the drama club, which is $\(|S \backslash D|\)$.

This can also be written as $\(|S|-|D \cap S|\)$, because:
- Total students union members = members in both + members only in students union.
- So, members only in students union = total students union - members in both.

Therefore:

$$
\(|S \backslash D|=90-|D \cap S|\)
$$


Finding the Range for $\(|S \backslash D|\)$
We know $\(|D \cap S| \leq 70\)$. Let's see what this implies for $|S \backslash D|$ :

$$
\(|S \backslash D|=90-|D \cap S| \geq 90-70=20\)
$$


So, $\(|S \backslash D| \geq 20\)$.
But can $\(|D \cap S|\)$ be less than 70 ? Yes, it can be any number from 0 up to 70 .
- If $\(|D \cap S|=70\)$, then $|S \backslash D|=20$.
- If $\(|D \cap S|\)$ decreases, $\(|S \backslash D|\)$ increases.

The maximum $\(|D \cap S|\)$ can be is 70 (as given), and the minimum is theoretically 0 (if no drama club members are in the students union), but we also have to consider that the drama club has 75 members, and the students union has 90 , so the overlap can't be such that it violates these numbers.

But since $\(|D|=75$ and $|S|=90\)$, the maximum overlap is the smaller of $|D|$ and $|S|$, which is 75 . But we're told that at most 70 are in both, so the overlap is capped at 70.

Similarly, the minimum overlap is when as many drama club members as possible are not in the students union. Since there are 75 drama club members and 90 students union members, the minimum overlap is:

$$
\(|D \cap S| \geq|D|+|S|-|D \cup S|\)
$$


But $\(|D \cup S| \leq\)$ total students, which isn't given, so this might not help directly. Alternatively, since the students union has 90 members, and the drama club has 75 , the smallest overlap occurs when as many drama club members as possible are outside the students union.

But since the drama club has 75 members, and at most 70 are in the students union, the minimum $\(|D \cap S|\)$ is when the least possible are in both.

Wait, the problem states "at most 70 of the drama club members are in the students union," which means $\(|D \cap S| \leq 70\)$. It doesn't say a minimum, so $\(|D \cap S|\)$ can be as low as 0 .

But if $\(|D \cap S|=0\)$, then all drama club members are not in the students union, which is possible.
Then:

$$
\(|S \backslash D|=90-0=90\)
$$


But we have to check if this is possible given the numbers.
Total students union: 90. If none of the drama club members are in it, then the number not in drama club is 90 , which is possible (since drama club is separate in this case).

Similarly, if $\(|D \cap S|=70\)$, then $\(|S \backslash D|=20\)$.
So $\(|S \backslash D|\)$ can range from 20 up to 90 .
But let's see the options given: A) $5, B) 15, C) 20, D) 75, E) 90$.
From our reasoning, possible values are $\(\geq 20\)$, so possible options are $C, D, E$.
But let's double-check if lower values are possible.
Can $\(|S \backslash D|\)$ be less than 20? That would require $|D \cap S|>70$, but the problem says at most 70, so no. So minimum is 20 .

Thus, possible options are C (20), D (75), E (90).
But let's check if 75 is possible.
For $\(|S \backslash D|=75\)$ :

$$
\(\begin{gathered}
75=90-|D \cap S| \\
|D \cap S|=15
\end{gathered}\)
$$


Is this allowed? Yes, because $\(|D \cap S| \leq 70\)$, and $\(15 \leq 70\)$.
Similarly, for $\(|S \backslash D|=75\)$, it's possible if 15 drama club members are in the students union.
Similarly, for $\(|S \backslash D|=90,|D \cap S|=0\)$, which is allowed.
For $\(|S \backslash D|=20,|D \cap S|=70\)$, which is the maximum allowed, so also fine.
Now, can $\(|S \backslash D|\)$ be between these values? Like 25, 30, etc.? Yes, but since the options are specific, we only consider given ones.

So possible are C, D, E.