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 $n^k$
• 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:

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: