We found out that we have and choose anyone possibilities or number of arrangements inside the box.
The first box inside the first energy level, and one can be any number.
But please keep in mind, obviously should be obvious, but we know that.
And one plus and two plus the Dopplers and some M is equal to.
And this is a constraint.
And we just the variety does some over J.
And some J equals an.
The number of all the all the possible charges that they will not buy on W.
Is.
I mean, they are equal to this first.
I mean, the number of possible choices for the first box.
So ends up and one times the number of possibilities for the second box.
What is the number of possibilities for the second box?
Well, this is easy.
If I have I mean, it's not and choosing to, but there are N minus N one the remaining particle.
So I have to choose from and minus and one.
A number of particles equal to N2.
Right.
And then so on and so forth.
We have an A minus and one minus and two for the third one.
And we have to choose and three of them, dot, dot, dot until all the way until.
OK.
And minus some from a day ranging from one, two and minus one.
And sub J.
And then we have choose and some of them.
All right.
So this is the number of possible choices that we have if we expand the binomial coefficients.
We can rewrite it.
Uh, can we read this as W is equal to and factoid only so and choose.
And one is an factorial divided by N minus and one factorial and one factorial.
And then I have an minus and one choose and two which is an minus and one factorial divided by and minus
one minus and two factorial times and two for and two factorial.
And then for the third one, we have.
And minus one minus.
And two factorial divided by and minus one minus and two minus and three factorial times and three factorial.
Dot, dot, dot.
Let here please not let.
Let's just.
Just allow me to to stop here.
But take a look at the Deum expression.
So we have this part of the denominator which canceled the next one.
So this one cancels these one.
Just take a look.
Right.
So all these cross terms here, simplify this one will some will simplify with the one here and so on
and so forth.
So you get the idea because we have factorial divided by N1 factorial and two actonel and three factorial.
So this is just equal to end factorial divided by anyone subgoal and two factorial and three factorial
and so on and so forth, all the way up to end sub and factorial.
And this is the expression that we are going to work on.
In the next year, this is a very useful expression.