# The meaning of exactly one student when we assign each project to exactly one student

A teacher has 30 project titles to be assigned to 7 students. A project can be assigned to any non-negative number of students and each student can be assigned any non-negative number of projects. Find the number of ways of assigning project titles to students if

• A) There is no additional conditions
• B) Each student receives at least one project

• C) Each project is assigned to at least one student

• D) each project is assigned to exactly one student

• E) Each project is assigned to at exactly one student and each student receives at least one project

• F) Each project is assigned to at least one student and each student receives at least one project

• G) Each project is assigned to exactly two students and each student receives at least one project

• H) Each project is assigned to at most 5 student

When I try to solve this problem. I got the first 3 part ( A,B,C) but D part and E part says that each project is assigned to at EXACTLY one student. I think that this is impossible but I cannot be sure what the question means by saying exactly and then I got the F part. But G is again impossible I guess. For part H I couldnt do anything. I am trying to solve this problem and I need your help.

Shortly. I need your opinion for the part D,E,G and H part

• For a Part I said that the answer is $31^7$ because each student receives 1 to 30 different projects and they may take nothing. – Manzara Oct 22 '17 at 21:35
• @Jens then the answer for C is (2$^7$-1)$^3$$^0$ but I cannot solve the F part then both of them at least is very confused – Manzara Oct 22 '17 at 21:53