There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$.
1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$?
2) What is the number of students from club $A$?
1. Let \( p \) be the number of students who stand next to exactly one student from club \( A \) and one student from club \( B \).
2. Let \( q \) be the number of students who stand between two students from club \( A \).
3. Let \( r \) be the number of students who stand between two students from club \( B \).

From the problem, we have the following equations:
\[ p + q + r = 100 \]
\[ p + q = 62 \]
\[ p + r = 54 \]

We need to solve these equations to find the values of \( p \), \( q \), and \( r \).

First, subtract the second equation from the third equation:
\[ (p + r) - (p + q) = 54 - 62 \]
\[ r - q = -8 \]
\[ r = q - 8 \]

Now, substitute \( r = q - 8 \) into the first equation:
\[ p + q + (q - 8) = 100 \]
\[ p + 2q - 8 = 100 \]
\[ p + 2q = 108 \]

Next, subtract the second equation from this result:
\[ (p + 2q) - (p + q) = 108 - 62 \]
\[ q = 46 \]

Now, substitute \( q = 46 \) back into the second equation:
\[ p + 46 = 62 \]
\[ p = 16 \]

Finally, substitute \( q = 46 \) into \( r = q - 8 \):
\[ r = 46 - 8 \]
\[ r = 38 \]

Thus, the number of students who stand next to exactly one student from club \( A \) and one student from club \( B \) is \( p = 16 \).

The final answer is \( \boxed{16} \).