AMC 10B 2024 (Problem 12)

A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students

A

and

B

, student

A

speaks some language that student

B

does not speak, and student

B

speaks some language that student

A

does not speak. What is the least possible total number of languages spoken by all the students?

\text{(A)}\;9

\text{(B)}\;10

\text{(C)}\;12

\text{(D)}\;51

\text{(E)}\;100