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Now we can talk about factorial notation. One mathematical pattern that recurs frequently in counting problems is the product of some integer n with all the positive integers smaller than n. Because this situation occurs frequently, mathematicians have given the pattern its own name and notation. The pattern of n times all the positive integers less than n is called n factorial.
Thus 5 factorial means 5 times 4 times 3 times 2 times 1. Well of course that would be 20 times 6 or 120. The notation used for this is simply 5 followed by an exclamation point. We read that as 5 factorial. So, 1 factorial would be 1. 2 factorial would be 2 times 1, which is 2.
3 factorial, 3 times 2 times 1, would be 6. 4 factorial would be 4 times 6, which is 24. 5 factorial would be 5 times 24, which is 120. 6 factorial would be 6 times 120, which is 720. Notice we’re getting kind of big already. Factorials get very big, very quickly.
So I’ll just point out numbers you don’t need to know, but I’ll point out that 10 factorial is already more than a million. 13 factorial is already more than a billion. 15 factorial is already more than a trillion. Now, I’ll just say personally, for me, a trillion is where inconceivably big begins.
So 15 factorial is already inconceivably big. So no one is gonna ask you to evaluate something like 20 factorial. Really, there’s no calculator on Earth that could calculate 20 factorial. It’d just be too large of a number. But, the test could expect you to know the factors of 20 factorial and to recognize, for example, that 17 factorial or 13 factorial is a factor of 20 factorial.
So think about this, 20 factorial is 20 times 19 times 18 times 17 dot, dot, dot, all the way down to 3 times 2 times 1. So we could factor out the 20, and then leave everything else multiplied together and of course everything else that would be 19 times 18 times 17. So that would be 20 times 19 factorial. Now I could factor out a 19, leave all the other factors, so it’s 20 times 19 times 18 factorial.
Or I could factor out the 18, so now it’s 20 times 19 times 18 times 17 factorial. So it’s possible to unpack a factorial like this to see some of its factors. So it would be very easy, for example, to see what would happen if we divided 20 factorial by 19, or something like that. So far, in counting problems, we have learned that if we have n different items, we can put these items in n factorial different orders.
We will make extensive use of the factorial notation in the next few lessons. In summary, that notation, n followed by an exclamation point, which is read n factorial, is the product of n and all the positive integers less than n. Any factorial is divisible by all the integers less than n and all the factorials less than n.
So for example, 30 factorial has to be divisible by every number less 30. It also has to be divisible by every factorial less than 30, and n different items can be arranged in n factorial unique orders. That’s the beginning of the use of the idea of factorials in combinatorics, and again, we’re gonna be using it quite extensively in the upcoming lessons.
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