Consider a a single large two$-$state paramagnet that consists of $N$ elementary dipoles.

$($a$)$ Use Stirling's approximation for factorials $N{N}^N{e}^{-N}\sqrt[\phantom{2}]{2N}$ to find a $($very compact$)$ formula for the probability of having exactly half the dipoles pointing up and half pointing down. Be sure to simplify your formula as much as possible.

Evaluate your approximate formula for this probability numerically $($to at least $7$ decimal places$)$ for $N=100,N=1000,N=10000.$

$($b$)$ Use Mathematica $($from AppsAnywhere, or use any other Maths software$)$ to find the "exact" answer for this probability $($to $7$ d$.$p$.)$ using $N!,$ for $N=100,N=1000,N=10000.$

Compare to your answers to part $($a$).$

Your normal calculator might be out of its depth for this.

$($The Mathematica function for combinations is called Binomial.$)$

$($c$)$ For $N=100$ dipoles, plot a graph $($with Mathematica or any other graph plotting software$)$ of the probability as a function of the number of dipoles pointing up $)=(0,..,N.$

What is the total area under the graph?

$$ The total number of all possible outcomes is: $($number of choices$)^$number $of$ objects

$$ If we are choosing $n$ out of $N,$ then the number of combinations is: $C=\frac{N!}{n!(N-n)!}.$

$$ The probability $p$ is the number of combinations divided by the total number of outcomes. It is a number between $0$ and $1.$