There are $S$ students, $I$ interviewers, each student has to undergo $R$ interviews, and each interviewer can interview at most $X$ students. No student interviews with an interviewer more than once, and no interviewer interviews a student more than once.
Find the number of possible student interviewer assignments.
If we suppose any interviewer can choose $K_{i}$ student , and $\begin{equation}\ 0 \leq K_{i} \leq X \end{equation}$ student then all of interviewers must pick all the student so we have :$$K_{1}+K_{2}+K_{3}+K_{4}+...+K_{i}=S ,\begin{equation}\ 0 \leq K_{i} \leq X \end{equation}$$
so we have : $\begin{equation}\ 0 \leq X-K_{i} \ \end{equation}$ and with $t=X-K_{i}$
we have : $$t_{1}+t_{2}+t_{3}+t_{4}+...+t_{i}=IX-S ,\begin{equation}\ 0 \leq t_{i} \ \end{equation}$$
then the number of possible student interviewer assignments is : $$\binom{IX-S+I-1}{I}$$