Combinatorics. I did my task, but not sure it is right.

[Olay]

Please help me...
A teacher has 16 tasks. He has nine students who can help. In how many ways can the teacher gives tasks to students if each student must get at least one task to avoid being out of work? There is no need to discriminate between tasks, only the number of tasks is important.
C(n,k)=n!/(n-k)!k!
C(16,9) = 16!/(16-9)!9! = 11440
Am I right?
Thank you

 

[AmandasMathHelp]

This question is definitely more complicated than just doing 16 choose 9. It might help to notice that each student needs at least one task, so you can assume that each student is given one task right off the bat. Then there are 7 tasks left, and unfortunately they can be distributed a lot of different ways. Maybe one student gets all 7 and the other 8 get no more tasks. I will represent that like this 7,0,0,0,0,0,0,0,0. Or maybe one gets 6 and one of the other students get another 6,0,0,0,1,0,0,0,0 (there are a couple of options there). Or maybe they are distributed more like this. 0,2,1,0,1,2,1,0,0. There are A LOT of different possibilities. So you need to think about this a little more deeply to come up with an answer. Good luck!

 

[Dr.Peterson]

Please help me...
A teacher has 16 tasks. He has nine students who can help. In how many ways can the teacher gives tasks to students if each student must get at least one task to avoid being out of work? There is no need to discriminate between tasks, only the number of tasks is important.
C(n,k)=n!/(n-k)!k!
C(16,9) = 16!/(16-9)!9! = 11440
Am I right?
Thank you

No, that's the number of ways to choose 9 of the tasks to be done; it has nothing to do with how many each student does.

What you want is the number of ways to put a positive integer in each of these 9 slots, _ _ _ _ _ _ _ _ _, so that the total is 16.

What have you learned about combinatorics? Anything beyond basic combinations and permutations?

 

[pka]

A teacher has 16 tasks. He has nine students who can help. In how many ways can the teacher gives tasks to students if each student must get at least one task to avoid being out of work? There is no need to discriminate between tasks, only the number of tasks is important.

Give one task to each student, then: If that means that the tasks are essentially the same, then there are \(\dfrac{15!}{7!\cdot 8!}=6435\) ways to place seven identical objects into nine distinct cells.

 

[Olay]

Give one task to each student, then: If that means that the tasks are essentially the same, then there are \(\dfrac{15!}{7!\cdot 8!}=6435\) ways to place seven identical objects into nine distinct cells.

Thank you for your help!