Question

number of ways in which 4 basons can be distributed in 4 states

Answer 1

get 165 using an equation I derived (which is correct), but when I do it the manual way I'm short by 36... This question is too much to ask of you to do the manual way, particularly on an exam.

DISCLAIMER: LONG ANSWER! (obviously)

METHOD ONE: DRAWING OUT MACROSTATES

I'm going to do this a bit out of order, because having one particle per box is the most confusing one.

1) All particles in the same box

Here is the representative macrostate:



With 9 boxes, you get 9 ways to have three-particle boxes.

3) Different boxes in the same row

Here is the representative macrostate:



With two particles in the same box, you have 6 configurations of the remaining particle in its own box within the same row. With 3 rows, that gives 18 configurations.

4) Different boxes in the same column

Here is the representative macrostate:



Rotate the box 90∘, then repeat (3) for 18 more, except we would have technically done it column-wise because the boxes are distinguishable.

5) Different boxes in different rows AND columns (at the same time)

Here is the representative macrostate:



You should get 4 configurations with one two-particle box in the upper left, times three columns for the two-particle box within the same row equals 12 configurations. Multiply by the three rows to get 36 configurations.

2i) Each particle in its own box, row-wise

Play around with this. You should get:

a) All particles in the same row: 3 configurations

Here is the representative macrostate:



b) Two particles in the same row: 2 configurations if two particles are in the first two columns, 2 configurations if the particles are in columns 1 and 3, 2 configurations if the particles are in the last two columns, times 3 rows, for 18 total.

Here is the representative macrostate:



c) All particles in different rows: 2 diagonal configurations, 4 configurations with two particles on the off-diagonal, for a total of 6.

Here is the representative macrostate:



Apparently, I get 3+18+6=27 here.

2ii) Each particle in its own box, column-wise

Play around with this. You should get:

a) All particles in the same column: 3 configurations

Here is the representative macrostate:



b) Two particles in the same column: 2 configurations if two particles are in the first two rows, 2 configurations if the particles are in rows 1 and 3, 2 configurations if the particles are in the last two rows, multiplied by the 3 columns, for 18 total.

Here is the representative macrostate:



c) All particles in different columns: We don't count these, because they are redundant.

Apparently, I get 

Answer 2

Answer:

35

Explanation: