Permutations

Permutations (Ordered Arrangements)

An arrangement (or ordering) of a set of objects is called a permutation. (We can also arrange just part of the set of objects.)

In a permutation, the order that we arrange the objects in is important

Example 1

Consider arranging 3 letters: A, B, C. How many ways can this be done?

ANSWER

The possible permutations are

ABC, ACB,
BAC, BCA,
CAB, CBA.

Hence, there are six distinct arrangements.

Another way of looking at this question is by drawing 3 boxes.

Any one of the A, B, C goes into the first box (3 ways to do this), and then the remaining one of the two letters goes into the second box (2 ways to do this), and the last remaining letter goes into the third box (only one way left to do this).

Hence, total no of ways $\displaystyle{3}\times{2}\times{1}={6}$

Reminder - Factorial Notation

Recall from the Factorial section that n factorial (written $\displaystyle{n}!$) is defined as:

n! = n × (n − 1) × (n − 2) ... 3 × 2 × 1

Each of the theorems in this section use factorial notation.

Theorem 1 - Arranging n Objects

In general, n distinct objects can be arranged in $\displaystyle{n}!$ ways.

Example 2

In how many ways can $\displaystyle{4}$ different resistors be arranged in series?

ANSWER

Since there are $\displaystyle{4}$ objects, the number of ways is

$\displaystyle{4}!={24}$ ways

Theorem 2 - Number of Permutations

The number of permutations of n distinct objects taken r at a time, denoted by $\displaystyle{{P}_{{r}}^{{n}}}$, where repetitions are not allowed, is given by

$\displaystyle{{P}_{{r}}^{{n}}}={n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}\ldots{\left({n}-{r}+{1}\right)}=\frac{{{n}!}}{{{\left({n}-{r}\right)}!}}$

Notes

(1) $\displaystyle{{P}_{{n}}^{{n}}}={n}!$ (since $\displaystyle{0}!={1}$)
(2) Some books use the following notation for the number of permutations:

$\displaystyle{n}{P}{r}$

and others have:

$\displaystyle{}^{n}{P}_{{r}}$

Example 3

In how many ways can a supermarket manager display $\displaystyle{5}$ brands of cereals in $\displaystyle{3}$ spaces on a shelf?

ANSWER

This is asking for the number of permutations, since we don't want repetitions. The number of ways is:

$\displaystyle{{P}_{{3}}^{{5}}}=\frac{{{5}!}}{{{\left({5}-{3}\right)}!}}$ $\displaystyle=\frac{{{5}!}}{{{2}!}}$ $\displaystyle={60}$

Example 4

How many different number-plates for cars can be made if each number-plate contains four of the digits $\displaystyle{0}$ to $\displaystyle{9}$ followed by a letter A to Z, assuming that

(a) no repetition of digits is allowed?

(b) repetition of digits is allowed?

ANSWER

(a) There are $\displaystyle{10}$ possible digits $\displaystyle{\left({0},{1},{2},\ldots,{9}\right)}$ and we need to take them $\displaystyle{4}$ at a time. There are $\displaystyle{26}$ letters in the alphabet.

With no repetition, we have:

$\displaystyle{{P}_{{4}}^{{10}}}\times{26}=\frac{{{10}!}}{{{\left({10}-{4}\right)}!}}\times{26}$ $\displaystyle=\frac{{{10}!}}{{{6}!}}\times{26}$ $\displaystyle={131040}$

(b) With repetition, we have simply:

(number of digits from $\displaystyle{0000}$ to $\displaystyle{9999}$)$\displaystyle\times{26}$

$\displaystyle={10},{000}\times{26}$
$\displaystyle={260},{000}$

Theorem 3 - Permutations of Different Kinds of Objects

The number of different permutations of n objects of which n₁ are of one kind, n₂ are of a second kind, ... n_k are of a k-th kind is

$\displaystyle\frac{{{n}!}}{{{n}_{{1}}!\times{n}_{{2}}!\times{n}_{{3}}\times\ldots\times{n}_{{k}}!}}$

Example 5

In how many ways can the six letters of the word "mammal" be arranged in a row?

ANSWER

We use Theorem 3.

Since there are three "m"s, two "a"s and one "L" in the word "mammal", we have for the number of ways we can arrange the letters in the word "mammal":

$\displaystyle\frac{{{6}!}}{{{3}!\times{2}!\times{1}!}}={60}$

Theorem 4 - Arranging Objects in a Circle

There are $\displaystyle{\left({n}-{1}\right)}!$ ways to arrange n distinct objects in a circle (where the clockwise and anti-clockwise arrangements are regarded as distinct.)

Example 6

In how many ways can $\displaystyle{5}$ people be arranged in a circle?

ANSWER

$(5-1)!=4!=24$ ways