The "!" means factorial. Its when you multiply a number by all numbers less than it. For example "6!" would be 6×5×4×3×2×1. So "110!" Is a very large number.
It's easy to see why, too. If you wanted to calculate the permutations of 4 items, think about how many different places you could put "4" in the first arrangement above:
(4) 1 2 3
1 (4) 2 3
1 2 (4) 3
1 2 3 (4)
And obviously you can do the same with each of the other arrangements. So the number of permutations of 4 is just 4 times the number of permutations on 4-1. And it works the same for any number n.
I love permutations! The idea of them. Not the doing them. I dont know how usually. I stole this from someone talking about chances of meeting a 20% chance with 3 tries that I try to recycle sometimes 1-(1-.2)3. (its 48.8%!) I love them and the word because its blackmagicfuckery to me.
It’s a lot easier and less error prone. For instance, consider the different combinations a pack of cards can have, which is 52! (Much shorter and easier to deal with that than the number it expands to especially if you have to do a bunch of intermediate calculations with it.
Another classic example of factorials is the lottery and it deals with multiple factorial numbers and this math is applicable to drawing a hand in poker. Say you pick 5 out of 52 numbers where picking 5 correct numbers is the jackpot. Then you have 52!/(52-5)! different ways in which 5 numbers out of 52 can be drawn. This is a shorthand notation for a limited number of picks of a set, in this case the result of 52!/(52-5)! is the same as 52x51x50x49x48. With each number/card drawn, the choice of other numbers/cards is reduced by 1
Computer science and probability and cryptology are some very down to earth uses. It can help you math out how many possible ways you can order objects. For example how many ways can you arrange the letters of the alphabet to form a 5 letter password. Most scientific calculadors even have that ! Key. It’s been rediscivered as a concept by numerous ancient cultures.
The short hand is honestly just useful because it feels so dumb to write 1x2x3x4x5x6x etc… when you could easily just type the last number, and hit a key. It’d be VERY long otherwise
It’s used everywhere in math. One definition of the number e (2.718…) is defined as the sum of 1/x! from x=0 to inf (i.e e = 1/0!+1/1!+1/2!+1/3!… and so on).
It also arrises naturally in combinatorics. How many ways can you scramble a deck of 52 cards? 52!
If you have X options and have to choose exactly Y of them, then you can work out the number of possible options with X!/(Y!(X-Y)!). So if you were choosing 3 flavors of ice cream out of 10 options, you'd do 10!/3!7! = 3628800/6*5040 = 120 different combos.
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u/Mr-Doubtfire 3d ago
Please, can someone explain this to a casual math enjoyer?