How Do You Calculate Microstates for a Constrained Spin System?

[Nylex]

Consider 2 systems of spin 1/2 paramagnets, which may point either up or down wrt a magnetic field. The first system contains 8 paramagnets and the second contains 6 paramagnets.

Suppose the energy of the combined system is constrained such that the total number of spins pointing up in the 2 systems is forced to be 6. Make a table of the possible values of n (no. of paramagnets pointing up, in 1st system) and m (no. of paramagnets pointing up, in 2nd system) and the number of microstates of the combined system for each case. By summing the entries of your table, obtain the total number of accessible microstates subject to the energy constraints.

This is what I've done so far:

m | n | accessible microstates
6 | 0 | (6C6) x (8C0) = 1
5 | 1 | (6C5) x (8C1) = 48
4 | 2 | (6C4) x (8C2) = 420
3 | 3 | (6C3) x (8C3) = 1120
2 | 4 | (6C2) x (8C4) = 1050
1 | 5 | (6C1) x (8C5) = 336
0 | 6 | (6C0) x (8C6) = 28

Total no. of accessible microstates = 3003.

I'm not sure whether or not my method for working out the no. of accessible microstates is right.

 

[wais]

Can someone check it for me?

Your method for calculating the number of accessible microstates is correct. The formula for calculating the number of microstates for a given system is given by the binomial coefficient, which you have correctly used in your table. The total number of accessible microstates for the combined system is obtained by summing the entries in your table, which gives the correct answer of 3003. Good job!