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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Q: How many members of the U.S.S. Enterprise does it take to change a
light bulb?
A: Seven. Scotty has to report to Captain Kirk that the light bulb in
the Engineering Section is getting dim, at which point Kirk will send
Bones to pronounce the bulb dead (although he'll immediately claim
that he's a doctor, not an electrician). Scotty, after checking
around, realizes that they have no more new light bulbs, and complains
that he "canna" see in the dark. Kirk will make an emergency stop at
the next uncharted planet, Alpha Regula IV, to procure a light bulb
from the natives, who, are friendly, but seem to be hiding something.
Kirk, Spock, Bones, Yeoman Rand and two red shirt security officers
beam down to the planet, where the two security officers are promply
killed by the natives, and the rest of the landing party is captured.
As something begins to develop between the Captain and Yeoman Rand,
Scotty, back in orbit, is attacked by a Klingon destroyer and must
warp out of orbit. Although badly outgunned, he cripples the Klingon
and races back to the planet in order to rescue Kirk et. al. who have
just saved the natives' from an awful fate and, as a reward, been
given all light bulbs they can carry. The new bulb is then inserted
and the Enterprise continues on its five year mission.