Arrangement
What is an Arrangement in combinatorics ?
An arrangement of p elements from n elements is an ordered selection of p elements from n elements.
Examples:
- {2, 3, 1} is an arrangement of 3 elements from {1, 2, 3, 4}.
- {c, a} is an arrangement of 2 elements from {a, b, c}.
Arrangement calculation Formula
Assume we have a set of n different objects then, the number of arrangements of p objects from n is equal to,
`A_n^p = frac {n!}{(n-p)!}`
Example: E is a set of 3 digits 1, 2, 3.
The number of arrangements of 2 elements from these 3 elements set is equal to 3! /(3-2)! = 6.
These arrangements are:
1, 2,
1, 3,
2, 1,
2, 3,
3, 1,
3, 2.
How to identify an arrangement ?
there are two important criteria in identifying an arrangement:
- Order matters
- Only a subset of the whole set is concerned
Example: how many possible trifecta bets are they in a 12 horse race ?
This is an arrangement because the order (of arrival) matters but, only a subset of horses is concerned (3 of 12).
The number of possible trifecta bets is 12!/(12-3)! = 1320
Comparison of enumeration methods
| Method | Elements concerned | Order matters ? | Formula |
|---|---|---|---|
| Permutation | All elements (n) | Yes | `n!` |
| Arrangement | Subset of p elements from n | Yes | `frac {n!}{(n-p)!}` |
| Combination | Subset of p elements from n | No | `frac {n!}{p!* (n-p)!}` |