Summary: Permutation Arrangement Combination

• 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.
• With repetition \(n^k\)
• No repetition \(P^n_k=\frac{n!}{ (n-k)!} = \binom{n}{k} \ . \ !k\)   $P^n_k = n \times (n-1) \times (n-2) \times \cdots \times (n-(k-1)) $
• Unordered => "Combination"
  Bag of stuff: pick \(k\) elements from \(n\) bins/objects.
• With repetitions \(\binom{n+k-1}{k}\)
• No repetitions \(C^n_k =\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Permutation (no repetitions / duplicates)

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

example:

3! = 6

123　　
321
213
231
132
312

Or with colors:

Combination (no repetitions / duplicates)

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

We pick 3 (order does not matter):

$$ {4 \choose 3 } = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4!}{3!} = \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} = 4 $$

Other kind of dispositions:

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Virtues vanishes in human interest like rivers in the sea.
(Fr: Les vertus se perdent dans l'intérêt comme les fleuves dans la mer.)
-- François de La Rochefoucauld (Réflexions ou Sentences et Maximes morales)