Summary: Permutation Arrangement Combination
There are 3 ways to make a disposition of objects:
Mnemonic: 3d PaC
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 $n^k$
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No repetition $P^n_k=\frac{n!}{ (n-k)!} = \binom{n}{k} \ . \ !k$
- 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:
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example:
3! = 6 123 321 213 231 132 312
Or with colors:
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$ {4 \choose 3 } $ From a box of 4:
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We pick 3 (order does not matter):
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$$ {4 \choose 3 } = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4!}{3!} = \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} = 4 $$
Note: Arrangement are often viewed as a sub-set of Permutations.
Other kind of dispositions:
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