zura wrote:
A professor will assign seven projects to three students. If two students each got 2 projects, and the other one got 3 projects, how many ways are possible?
A professor assigns three projects to seven students, therefore, the students will be divided two three groups, which has three, two, two students, respectively. How many ways are possible?
Responding to a pm:
Quote:
I believe the solution to the question should be 7C3 x 3C1 x 4C2 x 2C1.
However you have answered the question as 7C3 x 3C1 x 4C2. The reason I believe we should multiply it by 2C1 is because we need to select the student who gets the 2 projects selected by 4C2.
Your immediate & elaborate response would be most appreciated.
This is the reason your logic doesn't work:
The problem is with distributing 4 projects between 2 students such that each student gets 2 projects each.
Say students are Sa and Sb
Say projects are P1, P2, P3 and P4.
4C2 is the way you select 2 projects for Sa. Whatever is left is for Sb.
Say, you select
P1 and P2 for Sa. You have P3 and P4 for Sb.
P1 and P3 for Sa. You have P2 and P4 for Sb.
P1 and P4 for Sa. You have P2 and P3 for Sb.
P2 and P3 for Sa. You have P1 and P4 for Sb.
P2 and P4 for Sa. You have P1 and P3 for Sb.
P3 and P4 for Sa. You have P1 and P2 for Sb.
There are 6 ways which is 4C2.
Look at what happens when you do 4C2*2C1.
You
select P1 and P2 for Sa. You have P3 and P4 for Sb. Then since you multiply by 2,
you select P1 and P2 for Sb. You have P3 and P4 for Sa.In another case,
you select P3 and P4 for Sa. You have P1 and P2 for Sb. Then since you multiply by 2,
you select P3 and P4 for Sb. You have P1 and P2 for Sa.Note that you have double counted both cases. When you select 2 out of 4, you are selecting them for one particular person. So you do not multiply by 2C1.