A small seminar class consists of Anne, Bill, Cathy, Dan, Ellen, and Frank. The teacher has been told that 2 of them are from Indiana, 3 of them are from Michigan, and the other 1 is from Ohio. But the teacher does not know who is from which state. So the teacher decides to call on them one after another in alphabetical order and record which state each one is from until the teacher has found two (2) students from Indiana or has found a student from Michigan and a student from Ohio. How many overall outcomes are possible?
To find the number of overall outcomes, we need to consider the different possibilities for the order in which the students are called and the states that they are assigned to.
1. First, let's consider the order in which the students are called. Since the order is determined by alphabetical order, there is only one possible order for the students: Anne, Bill, Cathy, Dan, Ellen, Frank.
2. Next, let's consider the assignment of states to the students. We are given that 2 students are from Indiana, 3 students are from Michigan, and 1 student is from Ohio.
Case 1: Finding 2 students from Indiana first.
- The first student called could be from Indiana (Anne) or from Ohio (Bill, Cathy, Dan, Ellen, Frank).
- If the first student is from Indiana, the second student must also be from Indiana. There are two possibilities for the order in which the Indiana students are called: (Anne, Bill) or (Anne, Cathy).
- If the first student is from Ohio, the second student must be from Indiana. There are four possibilities for the order in which the Ohio and Indiana students are called: (Bill, Anne), (Cathy, Anne), (Dan, Anne), or (Ellen, Anne).
Case 2: Finding a student from Michigan and a student from Ohio first.
- The first student called could be from Michigan (Anne, Bill, or Cathy) or from Ohio (Dan, Ellen, or Frank).
- If the first student is from Michigan, we need to find a student from Ohio next. There are three possibilities for the order in which the Michigan and Ohio students are called: (Anne, Dan), (Bill, Dan), or (Cathy, Dan).
- If the first student is from Ohio, we need to find a student from Michigan next. There are three possibilities for the order in which the Ohio and Michigan students are called: (Dan, Anne), (Dan, Bill), or (Dan, Cathy).
Therefore, the number of overall outcomes is calculated as follows:
[Case 1: Finding 2 students from Indiana first] + [Case 2: Finding a student from Michigan and a student from Ohio first]
= (2 + 4) + (3 + 3)
= 12
Therefore, there are 12 overall outcomes possible in this scenario.