How many ways can the four students be assigned to the three sections, if each section must receive at least on new student?

Question

I got this question from the GRE book.

Central state college has three sections of Math 102. If four students, Bill, Jill, Phill, and Will decide to transfer from Math 101 into Math 102, in how many ways can the four students be assigned to the three sections, if each section must receive at least on new student?

A)12 B)18 C)24 D)27 E)36

I don't know how to solve this question, and I don't have a solution. Could you help me do this?

Thank you in advance.

Answer 1

It is clear that we need one section with two students, and two sections with one student. So, there are $3$ possibilities for what the section is with the two students, and once that is determined, there are ${4 \choose 2}=6$ ways to assign two students to that section. Finally, the remaining two students can be asigned to the remaining two section in $2$ different ways, and so the total number of possibilities is:

$$3 \cdot 6 \cdot 2=36$$

Answer 2

If there is no restriction that we must assign at least one new student to each section, then there are $3^4=81$ possibilities. (Consider the function from the set of students to the set of sections.)

Now we will rule out the possibility that there is a section with no new student. Consider the case to choose one section and put students into other two sections. There are $3$ sections and $2^4$ ways to assign students to remaining sections. Hence we must rule out $48=3\cdot 2^4$ cases...

No. We must consider there are double-counted cases. Can you find the cases which is double-counted?

Answer: The case when we assign all students to one section.

We must rull out these cases, and there are $3\cdot 1^4$ such cases.

Hence the answer is $3^4-(3\cdot 2^4 - 3\cdot 1^4) = 36$.

Answer 3

You're looking for the number of surjections from $\{1,2,3,4\}$ to $\{1,2,3\}$. Consider $A_i$ the set of functions such that $f^{-1}(\{i\})=\emptyset$. You want to compute $$|(A_1\cup A_2\cup A_3)^c|=3^4-|A_1\cup A_2\cup A_3|=3^4-(3\cdot 2^4-3\cdot 1+0)=36$$