# 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

Thanks for your help

Edit: I got the A,B,C,D and H part but I still dont get part e,f,g at least I need clue

Thanks for any help

• "Each project is assigned to exactly one student" means only one student will be assigned a given project. A student may be assigned more than one project but no project is assigned to more than one student. – Jens Oct 22 '17 at 20:26
• 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 part C and part D are the same ? – Manzara Oct 22 '17 at 21:36
• @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
• The answer to C cannot be the number you gave in the comment above as it is greater than the answer to A, which encompasses all possibilities. Regarding D, I ran a simulation which gave the answer 1,947,792. I'm reduced to doing simulations as the questions (at least to me) seem fairly complicated. – Jens Oct 22 '17 at 22:25