200 exams are to be graded by 7 teaching assistants, 5 of which are undergraduate and 2 are graduate TAs. TAs are distinct and assume the exams are distinct. Each of the undergraduate TAs should get at most 20 exams to grade. How many ways are there for the TAs to grade the exams?
Total exams = 200
Undergraduate TAs should get exams to grade =20
Graduate TAs should get exams to grade = 200 -20 = 180
Ways that undergraduate TAs grade the exam =7c5 * 20 =21 * 20 = 420
Ways that graduate TAs grade the exam =7c2 * 180 =21 * 180
= 3780
Total ways that TAs grade the exam = 7c5 * 20 + 7c2 * 180
=420 + 3780
= 4200
