Permutations

A permutation is an arrangement, or listing, of objects in which the order is important. In previous lessons, we looked at examples of the number of permutations of n things taken n at a time.

Permutation is used when we are counting without replacement and the order matters. If the order does not matter then we can use combinations.

The formula for permutation is

$$ P(n,r)=\frac {n!}{(n-r)!} $$

Example:

How many 3 letter words can we make with the letters in the word LOVE?

Solution:

There are 4 letters in the word love and making making 3 letter words is similar to arranging these 3 letters and order is important since LOV and VOL are different words because of the order of the same letters L, O and V.

Hence it is a permutation problem. The number of words is given by

$$ P(4,3)=\frac {4!}{(4-3)!} $$

$$ P(4,3)=24 $$

Factorial formula:

$$ n!= n \cdot (n − 1) \cdot (n − 2)\cdots 3 \cdot 2 \cdot 1 $$

Example:

Find the number of words, with or without meaning, that can be formed with the letters of the word ‘CHAIR’.

Solution:

‘CHAIR’ contains 5 letters.

Therefore, the number of words that can be formed with these 5 letters = 5!

$$ 5!=5\cdot 4\cdot 3\cdot 2\cdot 1 = 120. $$