The formal definition of #n!# (n factorial) is the product of all the natural numbers less than or equal to #n#. In math symbols:
#n! = n*(n-1)*(n-2)...#
Trust me, it's less confusing than it sounds. Say you wanted to find #5!#. You just multiply all the numbers less than or equal to #5# until you get to #1#:
#5! = 5*4*3*2*1=120#
Or #6!#:
#6! = 6*5*4*3*2*1=720#
The great thing about factorials is how easily you can simplify them. Let's say you're given the following problem:
Compute #(10!)/(9!)#.
Based on what I've told you above, you might think that you'll need to multiply #10*9*8*7...# and divide it by #9*8*7*6...#, which will probably take a long time. However, it doesn't have to be that hard. Since #10! = 10*9*8*7*6*5*4*3*2*1#, and #9! = 9*8*7*6*5*4*3*2*1#, you can express the problem like this:
#(10*9*8*7*6*5*4*3*2*1)/(9*8*7*6*5*4*3*2*1)#
And take a look at that! The numbers #1# through #9# cancel:
#(10*cancel9*cancel8*cancel7*cancel6*cancel5*cancel4*cancel3*cancel2*cancel1)/(cancel9*cancel8*cancel7*cancel6*cancel5*cancel4*cancel3*cancel2*cancel1)#
Leaving us with #10# as the result.
By the way, #0! = 1#. To find out why, check out this link .
Applications of Factorials
The place where factorials are really useful is probability. For example: how many words can you make from the letters #ABCDE#, without repeating any one letter? (The words in this case don't have to make sense - you can have #AEDCB#, for example).
Well, you have #5# choices for your first letter, #4# for your next letter (remember - no repetitions; if you chose #A# for your first letter, you can only choose #BCDE# for your second), #3# for the next, #2# for the one after that, and #1# for the last one. The rules of probability say the total number of words is the product of the choices:
#underbrace(5)_("choices for first letter")*4*3*2*1#
And four is the number of choices for the second letter, and so on. But wait - we recognize this, right! It's #5!#:
#5! = 5*4*3*2*1=120#
So there are #120# ways.
You'll also see factorials being used in permutations and combinations, which also have to do with probability. The symbol for permutations is #"_nP_r#, and the symbol for combinations is #"_nC_r# (people use #((n),(r))# for combinations most of the time, though, and you say "n choose r".) The formulas for them are:
#"_nP_r=(n!)/((n-r)!)#
#"_nC_r=(n!)/((n-r)!r!)#
There we see our friend, the factorial. An explanation of permutations and combinations would make this already long answer even longer, so check out this link for permutations and this link for combinations.
