Combinatorics Help: Splitting Dollar Notes and Functions with Sets M and N

[sintec]

I need a little help with combinatorics.

2 Students have 6 dollar notes worth 500 dollars, and 4 notes worth 1000 dollars. Notes with the same value are not distinguished.
A-How many ways to split the notes
B-How many ways to split the notes, so that both get an equal amount of notes.
C-How many ways to split the notes, so that both get an equal amount of money.
D- How many ways to split the notes if we distinguish the notes among each other.

And another:

There is given a set M with m elements and set N with n elements and function f:M->N
A- How many diferent bijective functions exist when m=n?
B- How many diferent injective functions exist when m<=n?
C- how many diferent surjective functions exist when m>=n?

thx!

 

[matt grime]

FOr the second one, as always, the answer is 'count'. Label the elements of M arbitrarily, the first can be sent to how many options? now take the second element, what do the restrictions on the type of function mean? so how many of the elements of N can it map to? rinse, repeat. that'll do a nd b, c needs you to think some more, but dont' give up. mathematics questions , if they jhave any merit, aren't supposed to be solved immediately.

 

[tacman]

A- There are 10 different ways to split the notes: 6 notes worth 500 dollars each, 4 notes worth 1000 dollars each, 5 notes worth 500 dollars and 1 note worth 1000 dollars, 4 notes worth 500 dollars and 2 notes worth 1000 dollars, 3 notes worth 500 dollars and 3 notes worth 1000 dollars, 2 notes worth 500 dollars and 4 notes worth 1000 dollars, 1 note worth 500 dollars and 5 notes worth 1000 dollars, 6 notes worth 1000 dollars, 5 notes worth 1000 dollars and 1 note worth 500 dollars, 4 notes worth 1000 dollars and 2 notes worth 500 dollars, and 3 notes worth 1000 dollars and 3 notes worth 500 dollars.

B- There are 4 different ways to split the notes equally: both students get 3 notes worth 500 dollars each, both get 2 notes worth 500 dollars and 2 notes worth 1000 dollars, both get 1 note worth 500 dollars and 3 notes worth 1000 dollars, or both get 4 notes worth 1000 dollars each.

C- There is only 1 way to split the notes so that both students get an equal amount of money: both get 5 notes worth 500 dollars and 2 notes worth 1000 dollars.

D- If we distinguish the notes among each other, there are 10!/(6!x4!) = 210 different ways to split the notes.

For the second question:

A- When m=n, there are n! different bijective functions.

B- When m<=n, there are n!/(n-m)! different injective functions.

C- When m>=n, there are m^n different surjective functions.