Combinatorics: 10 scarfs, 8 shirts and 9 hats to 4 people

[Iva]

Hello. I've been trying to solve these two problems for days now and it obviously didn't work out . All the help I found online so far was related to a theorem called stars and bars which seems to work. It felt weird to me because we haven't learned about it in class so I went to ask my professor if it's okay for me to use it in solving these problems. All they told me was that I wouldn't get any credit for using it since that's not the point of the problems. They said that the first one is solved by using classic combinatorics and the second one needs to be solved by using a generic function. I couldn't find a way to implement these things in my problems so I would be very grateful if someone would help out. Here are the problems:
1) In how many ways can one divide 10 scarfs, 8 shirts and 9 hats to 4 people? (what I find confusing about this one is that there are no limitations given);
2) How many solutions are there for the equation x1+x2+x3+x4+x5+x6=40 where x1, x2, x3, x4, x5, x6 are odd natural numbers?

I'm sorry if I'm not posting in the right field of this site, it's my first time and this one seemed the closest to my problems.
Thank you in advance. <3

 

[Steven G]

Hello. I've been trying to solve these two problems for days now and it obviously didn't work out . All the help I found online so far was related to a theorem called stars and bars which seems to work. It felt weird to me because we haven't learned about it in class so I went to ask my professor if it's okay for me to use it in solving these problems. All they told me was that I wouldn't get any credit for using it since that's not the point of the problems. They said that the first one is solved by using classic combinatorics and the second one needs to be solved by using a generic function. I couldn't find a way to implement these things in my problems so I would be very grateful if someone would help out. Here are the problems:
1) In how many ways can one divide 10 scarfs, 8 shirts and 9 hats to 4 people? (what I find confusing about this one is that there are no limitations given);
2) How many solutions are there for the equation x1+x2+x3+x4+x5+x6=40 where x1, x2, x3, x4, x5, x6 are odd natural numbers?

I'm sorry if I'm not posting in the right field of this site, it's my first time and this one seemed the closest to my problems.
Thank you in advance. <3

This reminds me of the very 1st problem I ever solved on this forum
1)how many ways can you divide 10 scarfs between 4 people? imagine lining up the 10 scarfs, s1, s2, s3, ..., s10. How many ways can you partition this list of 10 items into fours groups (some may have 0). Think about how many ways you can put three line between the list of 10 scarfs. Like s1 s2| s3 s4 s5| s6 s7 s8| s9 s10.
Please answer this and then we'll help you go further.

 

[Iva]

This reminds me of the very 1st problem I ever solved on this forum
1)how many ways can you divide 10 scarfs between 4 people? imagine lining up the 10 scarfs, s1, s2, s3, ..., s10. How many ways can you partition this list of 10 items into fours groups (some may have 0). Think about how many ways you can put three line between the list of 10 scarfs. Like s1 s2| s3 s4 s5| s6 s7 s8| s9 s10.
Please answer this and then we'll help you go further.

From what you've told me I would assume that this is the solution. I also solved the second problem and I would like to ask for corrections if necessary

 

[Iva]

So looks like 1 person can get 'em all...right?

odd natural numbers are 1,3,5,7,......right?
so 1,1,1,1,1,35 is a solution, and 35 is highest possible variable....right?

Since this is all that is given in the problems, I would say yes to all of your questions.