A teacher wants to create groups of students working together or independently.
How many different subsets can she form? Use Polya's 4 step reasoning process to create groups containing 4, 3, 2, and 1 students then count how many subsets are available.
Choose a method to include ALL possibilities and show your work.
Students: { Abby, Ben, Chris, Dave }
Show all the possible subsets. Groups (subsets) should include all possible combinations of 1 student, 2 students, 3 students, and 4 students. A group of 0 students does not make sense in this situation.
No idea what "Polya's 4 step reasoning process" is, but
given 3 elements, we can form 2^4 subsets. This includes the null set, which we want to exclude.
so 2^4 - 1 = 15
There are 15 subsets possible.
I am sure you can list those, I will do the subsets consisting of 2 elements:
AB, AC, AD, BC, BD, CD
To determine how many different subsets can be formed, we can use Polya's 4-step reasoning process:
Step 1: Identify the elements or objects. In this case, the elements are the students: Abby, Ben, Chris, and Dave.
Step 2: Determine the group sizes. In this situation, we want to create groups containing 4, 3, 2, and 1 student.
Step 3: Apply the reasoning process to each group size.
a) For a group of 4 students, we choose all 4 students from the given set of 4. The number of ways to do this is determined by the combination formula: C(n, r) = n! / (r!(n-r)!). In this case, C(4, 4) = 4! / (4!(4-4)!) = 4! / (4!0!) = 1. Therefore, there is only 1 way to form a group of 4 students.
b) For a group of 3 students, we choose 3 students from the given set of 4. C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = 4. Therefore, there are 4 ways to form a group of 3 students.
c) For a group of 2 students, we choose 2 students from the given set of 4. C(4, 2) = 4! / (2!(4-2)!) = 4! / (2!2!) = 6. Therefore, there are 6 ways to form a group of 2 students.
d) For a group of 1 student, we choose 1 student from the given set of 4. C(4, 1) = 4! / (1!(4-1)!) = 4! / (1!3!) = 4. Therefore, there are 4 ways to form a group of 1 student.
Step 4: Calculate the total number of different subsets. To do this, we add up the number of ways for each group size. In this case, the total number of subsets is 1 + 4 + 6 + 4 = 15.
Therefore, the teacher can form a total of 15 different subsets of students, which include all possible combinations of 1 student, 2 students, 3 students, and 4 students.