Chapter 13, Problem 13A.2P

Interpretation Introduction

Interpretation:

A table for configurations $N=9$ for total energy $9\epsilon$ has to be drawn.  The most probable configuration has to be estimated without evaluating weights.  The weights have to be calculated and the most probable configuration have to be identified.

Concept introduction:

The population of the state is the average number of all the molecules occupying the distribution energy level.  The configuration is the number of ways a molecule can be distributed over the available energy state.  Weight of the configuration is the number of ways a configuration can be achieved.  The configuration with the highest value of weight is the most probable configuration.

Answer to Problem 13A.2P

A table for configurations $N=9$ for total energy $9\epsilon$ is drawn below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$
8	0	0	0	0	0	0	0	0	1
7	1	0	0	0	0	0	0	1	0
7	0	1	0	0	0	0	1	0	0
7	0	0	1	0	0	1	0	0	0
7	0	0	0	1	1	0	0	0	0
6	2	0	0	0	0	0	1	0	0
6	0	2	0	0	1	0	0	0	0
6	0	0	3	0	0	0	0	0	0
6	1	0	0	2	0	0	0	0	0
6	1	1	0	0	0	1	0	0	0
6	1	0	1	0	1	0	0	0	0
6	0	1	1	1	0	0	0	0	0
5	3	0	0	0	0	1	0	0	0
5	0	3	1	0	0	0	0	0	0
5	2	1	0	0	1	0	0	0	0
5	2	0	1	1	0	0	0	0	0
5	1	1	2	0	0	0	0	0	0
5	1	2	0	1	0	0	0	0	0
4	4	0	0	0	1	0	0	0	0
4	3	1	0	1	0	0	0	0	0
4	3	0	2	0	0	0	0	0	0
4	2	2	1	0	0	0	0	0	0
3	5	0	0	1	0	0	0	0	0
3	4	1	1	0	0	0	0	0	0
2	6	0	1	0	0	0	0	0	0
2	5	2	0	0	0	0	0	0	0
1	7	1	0	0	0	0	0	0	0
0	9	0	0	0	0	0	0	0	0

The most probable configuration estimated without evaluating weights is shown below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$
4	2	2	1	0	0	0	0	0	0

The weights of the various configurations are calculated in the table shown below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$	$W=\frac{N!}{N_0!N_1!N_2!N_3!N_4!N_5!N_6!N_7!N_8!N_9!}$
8	0	0	0	0	0	0	0	0	1	$\begin{array}{c}W=\frac{9!}{8!0!0!0!0!0!0!0!0!1!}\\ =\frac{9\times 8!}{8!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =9\end{array}$
7	1	0	0	0	0	0	0	1	0	$\begin{array}{c}W=\frac{9!}{7!1!0!0!0!0!0!0!1!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
7	0	1	0	0	0	0	1	0	0	$\begin{array}{c}W=\frac{9!}{7!0!0!0!0!0!0!1!0!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
7	0	0	1	0	0	1	0	0	0	$\begin{array}{c}W=\frac{9!}{7!0!0!1!0!0!1!0!0!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
7	0	0	0	1	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{7!0!0!0!1!1!0!0!0!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
6	2	0	0	0	0	0	1	0	0	$\begin{array}{c}W=\frac{9!}{6!2!0!0!0!0!0!1!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
6	0	2	0	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!0!2!0!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
6	0	0	3	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!0!0!3!0!0!0!1!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =84\end{array}$
6	1	0	0	2	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!1!0!0!2!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
6	1	1	0	0	0	1	0	0	0	$\begin{array}{c}W=\frac{9!}{6!1!1!0!0!0!1!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
6	1	0	1	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!1!0!1!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
6	0	1	1	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!0!1!1!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
5	3	0	0	0	0	1	0	0	0	$\begin{array}{c}W=\frac{9!}{5!3!0!0!0!0!1!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
5	0	3	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!0!3!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 1\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
5	2	1	0	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!2!1!0!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
5	2	0	1	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!2!0!1!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
5	1	1	2	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!1!1!2!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 1\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
5	1	2	0	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!1!2!0!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
4	4	0	0	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!4!0!0!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 4\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =630\end{array}$
4	3	1	0	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!3!1!0!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =2520\end{array}$
4	3	0	2	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!3!0!2!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 3\times 2\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1260\end{array}$
4	2	2	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!2!2!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 2\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =3780\end{array}$
3	5	0	0	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{3!5!0!0!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{3\times 2\times 5!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
3	4	1	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{3!4!1!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{3\times 2\times 4!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =2520\end{array}$
2	6	0	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{2!6!0!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{2\times 1\times 6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
2	5	2	0	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{2!5!2!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{2\times 1\times 5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =756\end{array}$
1	7	1	0	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{1!7!1!0!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7!}{1\times 7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
0	9	0	0	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{0!9!0!0!0!0!0!0!0!0!}\\ =\frac{9!}{1\times 9!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1\end{array}$

The most probable configuration is shown below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$
4	2	2	1	0	0	0	0	0	0

Explanation of Solution

The total energy of the sample is $9\epsilon$.

The total number of molecules in the sample is $9$.

A table for configurations for total energy $9\epsilon$ is drawn below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$
8	0	0	0	0	0	0	0	0	1
7	1	0	0	0	0	0	0	1	0
7	0	1	0	0	0	0	1	0	0
7	0	0	1	0	0	1	0	0	0
7	0	0	0	1	1	0	0	0	0
6	2	0	0	0	0	0	1	0	0
6	0	2	0	0	1	0	0	0	0
6	0	0	3	0	0	0	0	0	0
6	1	0	0	2	0	0	0	0	0
6	1	1	0	0	0	1	0	0	0
6	1	0	1	0	1	0	0	0	0
6	0	1	1	1	0	0	0	0	0
5	3	0	0	0	0	1	0	0	0
5	0	3	1	0	0	0	0	0	0
5	2	1	0	0	1	0	0	0	0
5	2	0	1	1	0	0	0	0	0
5	1	1	2	0	0	0	0	0	0
5	1	2	0	1	0	0	0	0	0
4	4	0	0	0	1	0	0	0	0
4	3	1	0	1	0	0	0	0	0
4	3	0	2	0	0	0	0	0	0
4	2	2	1	0	0	0	0	0	0
3	5	0	0	1	0	0	0	0	0
3	4	1	1	0	0	0	0	0	0
2	6	0	1	0	0	0	0	0	0
2	5	2	0	0	0	0	0	0	0
1	7	1	0	0	0	0	0	0	0
0	9	0	0	0	0	0	0	0	0

The most probable configuration can be estimated to be the one in which the population of the higher states is not more than the population of the previous state.  This criterion is fulfilled by the configuration given below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$
4	2	2	1	0	0	0	0	0	0

The expression to determine the weight of the configuration $\left(W\right)$ is shown below.

  $W=\frac{N!}{N_0!N_1!N_2!N_3!N_4!N_5!N_6!N_7!N_8!N_9!}$        (1)

Where,

• $N$ is the total number of objects in the system.
• $N_0,N_1,N_2\mathrm{...}$ are the number of objects in the ground state, in state $1$ and so on.

The value of $N$ is $9$.

The value of $N_0$ is $8$.

The value of $N_1$ is $0$.

The value of $N_2$ is $0$.

The value of $N_3$ is $0$.

The value of $N_4$ is $0$.

The value of $N_5$ is $0$.

The value of $N_6$ is $0$.

The value of $N_7$ is $0$.

The value of $N_8$ is $0$.

The value of $N_9$ is $1$.

Substitute the value of $N$ and $N_0,N_1,N_2,N_3,N_4,N_5,N_6,N_7,N_8,N_9$ in equation (1) as shown below.

  $\begin{array}{c}W=\frac{9!}{8!0!0!0!0!0!0!0!0!1!}\\ =\frac{9\times 8!}{8!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =\underset{\_}{9}\end{array}$

Similarly, the weights of the various configurations are calculated in the table shown below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$	$W=\frac{N!}{N_0!N_1!N_2!N_3!N_4!N_5!N_6!N_7!N_8!N_9!}$
8	0	0	0	0	0	0	0	0	1	$\begin{array}{c}W=\frac{9!}{8!0!0!0!0!0!0!0!0!1!}\\ =\frac{9\times 8!}{8!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =9\end{array}$
7	1	0	0	0	0	0	0	1	0	$\begin{array}{c}W=\frac{9!}{7!1!0!0!0!0!0!0!1!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
7	0	1	0	0	0	0	1	0	0	$\begin{array}{c}W=\frac{9!}{7!0!0!0!0!0!0!1!0!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
7	0	0	1	0	0	1	0	0	0	$\begin{array}{c}W=\frac{9!}{7!0!0!1!0!0!1!0!0!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
7	0	0	0	1	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{7!0!0!0!1!1!0!0!0!0!}\\ =\frac{9\times 8\times 7!}{7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
6	2	0	0	0	0	0	1	0	0	$\begin{array}{c}W=\frac{9!}{6!2!0!0!0!0!0!1!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
6	0	2	0	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!0!2!0!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
6	0	0	3	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!0!0!3!0!0!0!1!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =84\end{array}$
6	1	0	0	2	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!1!0!0!2!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
6	1	1	0	0	0	1	0	0	0	$\begin{array}{c}W=\frac{9!}{6!1!1!0!0!0!1!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
6	1	0	1	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!1!0!1!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
6	0	1	1	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{6!0!1!1!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
5	3	0	0	0	0	1	0	0	0	$\begin{array}{c}W=\frac{9!}{5!3!0!0!0!0!1!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
5	0	3	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!0!3!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 1\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
5	2	1	0	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!2!1!0!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
5	2	0	1	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!2!0!1!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
5	1	1	2	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!1!1!2!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 1\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
5	1	2	0	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{5!1!2!0!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1512\end{array}$
4	4	0	0	0	1	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!4!0!0!0!1!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 4\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =630\end{array}$
4	3	1	0	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!3!1!0!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 3\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =2520\end{array}$
4	3	0	2	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!3!0!2!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 3\times 2\times 1\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1260\end{array}$
4	2	2	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{4!2!2!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 2\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =3780\end{array}$
3	5	0	0	1	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{3!5!0!0!1!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{3\times 2\times 5!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =504\end{array}$
3	4	1	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{3!4!1!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4!}{3\times 2\times 4!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =2520\end{array}$
2	6	0	1	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{2!6!0!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6!}{2\times 1\times 6!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =252\end{array}$
2	5	2	0	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{2!5!2!1!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7\times 6\times 5!}{2\times 1\times 5!\times 2\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =756\end{array}$
1	7	1	0	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{1!7!1!0!0!0!0!0!0!0!}\\ =\frac{9\times 8\times 7!}{1\times 7!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =72\end{array}$
0	9	0	0	0	0	0	0	0	0	$\begin{array}{c}W=\frac{9!}{0!9!0!0!0!0!0!0!0!0!}\\ =\frac{9!}{1\times 9!\times 1\times 1\times 1\times 1\times 1\times 1\times 1\times 1}\\ =1\end{array}$

The most probable configuration is the one which has the highest weight.

Therefore, the most probable configurations are tabulated below.

$0\epsilon$	$1\epsilon$	$2\epsilon$	$3\epsilon$	$4\epsilon$	$5\epsilon$	$6\epsilon$	$7\epsilon$	$8\epsilon$	$9\epsilon$
4	2	2	1	0	0	0	0	0	0