Combinations

quote-controller-is-so-intuitive-even-your-mum-can-play-shigeru-miyamoto-70-80-01
Source: that Nintendo guy.

A Combination is any unordered subset of a larger set “S” (a bit different from a Permutation, innit?)

So if S={1,2,3}, then {1,2}={2,1} as order don′t count for squat round here.




Really though, that’s the only difference between the two (permutations and combinations, that is). Given this, how the hell do we know how many combos of r objects chosen from n total objects are there?



Well we know already that the number of permutations of “r” objects from “r” objects is r! (just use The Multiplication Principle  taking the number of ways to take “r” things from a set “r” big).

We also know that a particular combination (concerning “r” things, “r” being smaller than or equal to the total things, “n”) is any unordered subset of “r” objects from “n” total objects. (denoted ^n C_r pronounced “n choose r” or “choose r from n” (only plebs say this one)).

Whatever that number is (we don’t know yet, we’re gonna define it in a sec tho), because it’s a subset of “r” things that’s unordered, and we know that there are r! permutations of sets of this size (size “r”), we can then say that: ^n P_r = ^n C_r * r! -> ^n C_r  = \frac{^n P_r }{r!} = \frac{n!}{{(n-r)}!r!}

See, there you have the mathematical definition of a Combination!




Ahh, the beauty of figuring out what you don’t know through pretending you do know till you really do know.

Ya know?

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