# Summary: Permutation Arrangement Combination

Combinatorics cheat sheet - 05/2023 - #Jumble

**There are 3 ways to make a**

__disposition__of objects:__Mnemonic:__3d

**PaC**

**P**ermutation

**a**rrangement

**C**ombination

**Ordered =>****"Permutation"****Every object:**any possible ordered disposition of \(n\) objects.- With repetition \(n^n\)
- No repetition \(n!\)
**Subset of objects:****"Arrangement**" a disposition of \(k\) objects from a set of \(n\) objects.**Unordered => "**Combination"

Bag of stuff: pick \(k\) elements from \(n\) bins/objects.- With repetitions \(\binom{n+k-1}{k}\)
- No repetitions \(C^n_k =\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

In permutation we just switch object positions until every combination is exhausted:

example:

3! = 6 123 321 213 231 132 312

Or with colors:

$ {4 \choose 3 } $ From a box of 4:

We pick 3 (order does not matter):

$$ {4 \choose 3 } = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4!}{3!} = \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} = 4 $$__Note:__Arrangement are often viewed as a sub-set of Permutations.

Other kind of dispositions:

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