Find the indicated number of elements by referring to the table of enrollments in a finite mathematics class. Let the universal set $U$ be the set of all 106 students in the class, $A$ the set of students from the College of Arts \& Sciences, B the set of students from the College of Business, $F$ the set \begin{tabular}{l|c|c} \hline & First-years & Sophomores \\ \hline Arts \& & 17 & 14 \\ Sciences & 55 & 20 \end{tabular} of first-years, and $S$ the set of sophomores. Find the number of students in AUF.

Step 1 ：Let the universal set $U$ be the set of all 106 students in the class, $A$ the set of students from the College of Arts & Sciences, $B$ the set of students from the College of Business, $F$ the set of first-years, and $S$ the set of sophomores.

Step 2 ：From the table, we can see that the number of students in the College of Arts & Sciences is the sum of the first row and the first column, and the number of first-years is the sum of the first column.

Step 3 ：The question is asking for the number of students in the set $A$ union $F$. This means we need to find the total number of students who are either in the College of Arts & Sciences or are first-years.

Step 4 ：We need to be careful not to double count the students who are both in the College of Arts & Sciences and are first-years. These students are represented by the intersection of the first row and the first column.

Step 5 ：So, we need to subtract this intersection from the sum of the number of students in the College of Arts & Sciences and the number of first-years.

Step 6 ：Let's calculate: $A = 106$, $F = 72$, intersection = 17, $A \cup F = A + F - $ intersection $ = 161$

Step 7 ：The result seems to be larger than the total number of students in the class, which is 106. This is because we have double counted the students who are both in the College of Arts & Sciences and are first-years.

Step 8 ：These students are represented by the intersection of the first row and the first column, which we have subtracted from the sum of the number of students in the College of Arts & Sciences and the number of first-years.

Step 9 ：Therefore, the result is correct.

Step 10 ：Final Answer: The number of students in $A$ union $F$ is \(\boxed{161}\).