• 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:

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example:

3! = 6

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321
213
231
132
312

Or with colors:

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Combination (no repetitions / duplicates)

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

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We pick 3 (order does not matter):

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

Other kind of dispositions: