Permutation means arrangement of things in different ways.

**Permutations**

Permutation means arrangement of things in different ways. Let
us take 3 things A, B, C for an arrangement. Out of these three things two at a
time, we can arrange them in the following manner.

*AB* *AC* *BC*

*BA* *CA* *CB*

Here we find 6 arrangements. In these arrangements, order of
arrangement is considered. Note that the arrangement AB and the arrangement BA
are different.

The number of arrangements of the above is given as the number
of permutation of 3 things taken 2 at a time which gives the value 6.

This is written symbolically 3P_{2} = 6

Thus the number of arrangements that can be formed out of n
things taken r at a time is known as the number of permutation of n things
taken r at a time and is denoted as *nP _{r}*
or

We write nP_{r} as nP_{r} = n (n –1) ( n – 2) ….
[n – (r–1)]

The same nP_{r}
can be written in factorial notation as follows:

To find 10 *P*_{3},
Start with 10, write the product of 3 consecutive natural numbers in the
descending order]

In how many ways can five students stand for a photograph in a
row?

The number of ways in which 5 students can stand in a row is
same as the number of arrangements of 5 different things taken all at a time.

This can be done in 5*P*5
ways and 5*P*_{5} = 5! = 120 ways.

The number of permutations of n objects, where p_{1}
objects are of one kind, *p*_{2}
are of second kind … *p _{k}*
are of kth kind and the rest, if any, are of different kind is given by

Find the number of permutations of the letters in the word
‘STATISTICS’

Here there are 10 objects (letters) of which there are 3S, 3 T,
2 I and 1A and 1C

Therefore the required number of arrangements is

If
10 P_{r} = 720 find the value of r.

10 P_{r} = 720

10 P_{r} = 10 × 9 × 8

10 P_{r} = 10 P_{3}

r = 3

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11th Statistics : Chapter 7 : Mathematical Methods : Permutations |

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