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Statistics and Permutations

Statistics Assignment / Homework Help
Introduction


In our day to day life we have many choices for a certain task or commodity. Hence we always have a dilemma as to which to choose. To select the task or commodity of our choice, we have many different ways. This is the basic concept on which the concept of combination is built.

Sometimes we are concerned about the arrangement of things or order of things. For this purpose we use the method of permutations. Let us first discuss the concept of permutations.

Definition: different arrangements by taking all or some of the available choices for a thing or task is called permutations. If there are n number of different things and r is the required number of choices, then the number of permutations is denoted by nPr.

The formula for nPr is given by


The numerator of the formula is called n factorial and is denoted by n!. the denominator is (n-r) factorial and is denoted by (n-r)!. hence the formula becomes


Permutations is commonly used when arrangements of a thing or different things , people, alphabets, etc are concerned. In other words when order of things is important, we use permutations.

To understand the concept of permutations in more detail, let us consider the following examples.

1. How many 3 digit numbers can be formed using the digits 2,4,5,8,9 when no repetitions are allowed?


Solution:

we have 5 different digits out of which only 3 can be selected to form a 3-digit number. hence the number of different ways of doing this is


On simplifying we get 5P3=60.

Hence we can form 60 different 3-digit numbers using the given 5 digits.

Let us extend the question little more as follows.

Suppose we want to find how many of these numbers will be less than 500.

For a 3-digit number to be less than 500, the hundredth place must be filled by a number less than 5 namely either 2,4. Hence this can be done in 2 ways. After the hundredth place is taken by either 2 or 4, the remaining 2 places can be filled by either of the 4 digits. Hence this can be done in 4P2 ways.



=12.


To find the total number of numbers less than 500, we multiply 2*12=24. Hence 24 numbers are less than 500.

Now suppose we need to find the number of even numbers out of these.

For a number to be even, the unit places must be filled by an even number that is either 2,4 or 8. This can be filled in 3 ways. After filling the units place with either 2, 4 or 8, the remaining 2 places can be filled by the remaining 4 digits. This can be done in 4P2 ways. As calculated previously, we get 4P2=12.

Now the total number of even numbers is given by=3*12=36.

Let us look at another example where arrangement of books are dealt with.

In how many ways can we arrange 5 science, 2 English books and 7 maths books so that

a) The science books are together?


Solution:

since we want all the science books together, let us consider the science as one unit.

Now 2english+7maths=9books +1unit of science books=10 books can be arranged in 10! different ways.

Now consider only the science books. There are 5 science books which can be arranged in 5! different ways.

Hence the total number of ways in which we can arrange all the science books together is 10!*5!.

Since these two values are large, we leave the answer as that.

Suppose we want the English books together and maths books together.

Again consider the English books as one unit and maths books as one unit. Now 5scicne books+1unit of maths+1unit of English=7 books can be arranged in 7! Different ways.

There are 2 English books that can be arranged in 2! Ways and 7 maths books can be arranged in 7! Ways.

Hence total is 7!*2!*7!.

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