Montel's theorem

Okay, so imagine you have a bag of different colored marbles. Montel's theorem is a way to figure out how many different ways you can pick out a certain number of marbles from the bag, when the order you pick them doesn't matter.

Let's say you have 5 red marbles, 3 blue marbles, and 2 green marbles in the bag. If you want to pick out 3 marbles, Montel's theorem says you can figure out the answer by doing some math with factorials (which are just really big multiplication problems).

First, you write out the number of marbles you have for each color:

- Red: 5
- Blue: 3
- Green: 2

Then, you add up all those numbers to get the "total" number of marbles:

- Total: 5 + 3 + 2 = 10

Next, you take the factorial of the total number of marbles (which means you multiply all the numbers from that number down to 1):

- 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800

Then, you take the factorial of the number of marbles you want to pick out (in this case, 3):

- 3! = 3 x 2 x 1 = 6

Finally, you take the factorial of each color's number of marbles and multiply them together:

- Red: 5! = 5 x 4 x 3 x 2 x 1 = 120
- Blue: 3! = 3 x 2 x 1 = 6
- Green: 2! = 2 x 1 = 2

- 120 x 6 x 2 = 1,440

Then you divide the factorial of the total number of marbles by the product of the factorials of the numbers of marbles of each color and the factorial of the number of marbles you want to pick out:

- 10! / (5! x 3! x 2! x 3!) = 3,628,800 / 1,440 = 2,520

So, according to Montel's theorem, there are 2,520 different ways you can pick out 3 marbles from the bag, when the order you pick them doesn't matter. Pretty cool, huh?