The CSE$373/$ MAT$373$ final exam will ask you only True$/$False questions!$1$ There will be n $>=2$ questions divided into m in $[2,$ n$]$ categories with exactly ni $>0$ questions in category k in $[1,$ m$]($so$,$ n $=$ P$1>=$km nk$).$ Those n questions will be chosen by the TAs from a question bank containing N $(>=$ n$)$ questions. For i N $,$ the i$-$th question in the question bank has mi in $[1,$ m$]$ category numbers assigned to it meaning that if the question is included in the final exam it must be placed under exactly one of those mi categories in the exam.

Design an efficient algorithm that, given the parameters and the question bank described above, can tell the TAs if it is possible to choose n questions from the question bank under the constraints given above. If n questions can indeed be chosen, the algorithm must also find one such set of n questions.