 This video contains a proof of the factorial formula, which says that n choose x is equal to n factorial divided by x factorial times n minus x factorial. I'm going to do this by explaining why ten choose three is equal to ten factorial divided by three factorial times seven factorial. Okay, so suppose that we've got ten Scrabble tiles, three of which are marked with an s, and seven of which are marked with an f. Ten choose three is the number of different words that can be made with three s's and seven f's. So the question we need to answer is how many words can be made with these Scrabble tiles. Now there's an important subtlety that you need to understand from the very beginning. You can rearrange the tiles to make a new word, but not every rearrangement of the tiles produces a different word. For example, in this word here, I can take out the s's and put them back again in a different order. So this is a different rearrangement of the tiles, but it's not a new word. And of course I can also take out the f's and rearrange them. And I can produce another arrangement of the tiles which spells the same word. So when we're asking how many words can be made, we need to understand that the number of arrangements of the tiles is much greater than the number of words. In fact, the total number of arrangements of the tiles is the number of words times the number of arrangements per word. Or, put a different way, the number of words is equal to the total number of arrangements of the tiles divided by the number of arrangements per word. So in order to answer the question how many words can be made, we need to ask two questions. How many arrangements of the tiles are there? And secondly, how many arrangements are there per word? Let's deal with the first question first. How many arrangements of the tiles are there? So let's suppose that we're going to put the ten tiles down one at a time, going from left to right. For the first tile, there are ten options because you've got ten tiles to choose from. We can pick any of these tiles, but just for the sake of illustration, suppose that we pick f3. Next we've got to pick a second tile, but at this stage things are different because there are only nine tiles left. So there are only nine possibilities for the second tile. And this means that overall there are ten times nine possibilities for the first two tiles. Now there are only eight tiles left and only eight possibilities for the third tile. So for the first three tiles, the number of possibilities is ten times nine times eight. You can see that if we carry on in this way, there are seven possibilities for the fourth tile, six possibilities for the fifth tile, five possibilities for the sixth tile, four possibilities for the seventh tile, three possibilities for the eighth tile, two possibilities for the penultimate tile, and only one possibility for the last tile. So in total for all ten tiles, the number of possible ways of putting them down is ten times nine times eight times seven times six times five times four times three times two times one, which is ten factorial. The second question that we need to ask is how many arrangements of the tiles are there per word? So let's look at a word, this is the one that we constructed a moment ago, and see how many ways there are of rearranging the tiles to get the same word. Well what we can do is to take out the S's and put them back in again, seeing how many ways there are of doing this. Well, there are three tiles, so for the first tile that we put back in, there are three possibilities. For the next tile, there are only two possibilities, which gives us a total of three times two possibilities so far. There's obviously only one possibility for the last tile, so the total number of possibilities for taking out the S's and rearranging them is three times two times one. Next we need to take out the F's and see how many ways there are of putting those back into the word. To begin with we have seven F's, so there are seven possibilities for the first replacement. Now though there are only six F's left, so there are six possibilities for the second letter. Now there are five, so there are five possibilities for the third letter, and so on as before. You can see that the total number of ways of rearranging the letters to spell the same word is three times two times one times seven times six times five times four times three times two times one, which is three factorial times seven factorial. We've now shown that the total number of ways of arranging the tiles is ten factorial, and the number of rearrangements for each word is three factorial times seven factorial. This means that the total number of words is ten factorial over three factorial times seven factorial. But remember that the number of words we can make is ten choose three. So we've shown that ten choose three is equal to ten factorial divided by three factorial times seven factorial. Now it's clear that I could go through this explanation using the letters N and X in place of ten and three. And so this proves that N choose X is equal to N factorial divided by X factorial times N minus X factorial. Thank you for watching this video.