Permutation

The arrangement of a given number of items in a particular order. Not to be confused with combinations, which is similar but without consideration of order.

Provided:

• the number of distinct items to choose, $n$ (things you have); and
• the number of permutations, $r$ (‘slots’ to fill)

$${}^n{P}_r=n\times (n-1)\times ...\times (n-r+1)=\frac{n!}{(n-r)!}$$

When there are duplicates, they must be removed. In order to do so, the following formula is used:

$$\frac{r!}{r_{dup}!}$$

where:

• $r$ is the number of distinct items; and
• $r_{dup}$ is the number of duplicated items (one for each).

Where restrictions are applicable, it is important to deal with them first. Some common restrictions include:

• the use even and odd numbers; and
• the use of non-zero numbers.

Examples

The following are some examples of permutations. Note how order plays a role in each case respectively:

• How many 3-letter word arrangements can be formed from the letters “A”, “B”, “C”, “D”, and “E” in all possible orders with no repetition of letters?
  • 3-letter word: ‘slots’ to fill ($n$)
  • letters A to E: distinct items to choose ($r$)
  • all possible orders: order matters
• How many ways are there to arrange the letters in the word “ELEMENTAL” with no repetition of letters?
  • no repetition of letters: duplicating items need to be removed
  • “ELEMENTAL”: (three) repeated Es, (two) repeated Ls
• How many ways are there to choose 2 students from a class of 20 students, so as to make the first person a class representative, and the second a class treasurer?
  • first class representative, second class treasurer: order matters