AMC 10 B

Suppose the languages spoken are labeled $L_1, L_2, L_3, \dots, L_n$. Note that the collection of all subsets of $\{L_1, L_2, L_3, \dots, L_n\}$ of size $r$ will satisfy the conditions in the problem for any $r$ in the range $1 \le r < n$. For a given $n$, there are $\binom{n}{r}$ subsets, and $\binom{n}{r}$ is maximized when $r = \lfloor \frac{n}{2} \rfloor$ or $r = \lceil \frac{n}{2} \rceil$.

If $n = 8$ and $r = 4$, the number of distinct subsets is $\binom{8}{4} = 70 < 100$, so $n > 8$. But $\binom{9}{4} = \binom{9}{5} = 126 > 100$, so $n = 9$ is the least possible total number of languages spoken by all the students.