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Calling the system binary means that each magnet can be oriented either
in the "up" position or the "down" position, and no other. If a magnet
is in the down position, then we say that its magnetic moment is - m,
if up, it is + m. The magnets do not interact with each other; i.e.
the position of a magnet's neighbors does not influence its position. A
sample collection of such magnets can be seen in .

Magnetic moments add together just as vectors do. Therefore, we can
ask, how many ways are there to have a total magnetic moment M of
M = Nm? Such a state would require all of the magnets to be in the up
position, so there is only one way to achieve this state. How many ways
are there to have a total magnetic moment of M = (N - 2)m? Such a state
requires one magnet to be in the down position. Since there are N
magnets, there are N such ways.

Letting C represent the up position and D represent the down, we can
use a shorthand notation for representing all of the possible states of
the system:

(C + D)^{N}

Using a binomial expansion, and writing in summation
notation, we can write:

(C + D)^{N} = C^{N-i}D^{i}

The Multiplicity Function

Usually we are interested not in writing out a general form for all
states, but are more focused on one particular state. As we saw above,
sometimes there are multiple states with the same number of spins in the
up position. Let N_{up} be the number of particles in the "up" state,
and N_{down} be the number of particles in the "down" state
(then N = N_{up} + N_{down}). We refer to the number of states with the same
values of N and N_{up} by the function g(N, N_{up}), called the
multiplicity function. For our system, g(N, N_{up}) is given by the
coefficient in the preceding sum:

g(N, N_{up}) =

Notice that for very large and very small values of N_{up}, g is small,
but for N_{up} = N_{down}, g is a maximum.