A group of five students needs to break into two smaller groups in order to tackle the two different assignments associated with a larger group project. Naturally, they feel that it would be most fair to break into groups of two and three. How many ways are there to do this?
Choose one:
5
10
20
40
----------------------
My reasoning:
There are 5 students. I will name them student A, B, C, D, and E.
First grouping: AB
CDE
BC
DEA
etc. = 5 ways to combine students.
Second grouping: AC
BDE
BD
EAC
etc. = 5 ways to combine students.
Third grouping: CE
ABD
etc. = 5 ways to combine students.
Fourth grouping: AD
etc. = 5 ways
Fifth grouping: AE
etc. = 5 ways
5 groupings with 5 ways to combine students = 20 ways to group.
So, is the answer "20" or am I to assume
that they can be further grouped depending upon the unexplained tasks of
the class project?
Using combinations,
number of ways = C(5,3)xC(2,2) = 20x1 = 20
You are right.
Thank you!
You are correct in your reasoning that there are 5 ways to group the students for the first group and 5 ways to group them for the second group. However, just combining these two separate groups gives us the total number of ways to break into two smaller groups. It is not necessary to further group the students within each of the two smaller groups.
Therefore, the answer is the product of the number of ways to group students for the first group (5) and the number of ways to group them for the second group (5), which equals 25.
So, the correct answer is not among the options provided (5, 10, 20, 40).