Permutation and Combination

Permutations and Combinations

Permutations and Combinations‘ is the next post of my series Online Maths Tutoring. It is very useful and interesting as a topic. It’s also very useful in solving problems of Probability. To understand Permutations and Combinations, we first need to understand Factorial.

Definition of Factorial-

If we multiply n consecutive natural numbers together, then the product is called factorial of n. Its shown by n! or by

for example :       n! = n(n - 1)(n - 2)(n - 3)..........3.2.1

Some Properties of Factorials

(i) Factorials can only be calculated for positive integers at this level. We use gamma functions to define non-integer factorial that’s not required at this level

(ii) Factorial of a number can be written as a product of that number with the factorial of its predecessor    n! = n[(n - 1)(n - 2)(n - 3)..........3.2.1]

 = n(n - 1)!

(iii)  0! = 1  you can watch this video for the explanation.

(iv)  If we want to simplify a “permutations and combinations” expression that has factorials in the numerator as well as in the denominator, we make all the factorials equal to the smallest factorial

Exponent of Prime Number p in n!

Let’s assume that p is a prime number and n is a positive integer, then exponent of p in n! is denoted by  Ep (n!)

{E_p}(n!) = \left[ {\frac{n}{p}} \right] + \left[ {\frac{n}{{{p^2}}}} \right] + ........\left[ {\frac{n}{{{p^t}}}} \right]

We can’t use this result to find the exponent of composite numbers.

Fundamental Principle of Counting

Almost all IB Online Tutors, teach the first exercise of Permutations and Combinations that is based on the Fundamental Principle of Counting. We can learn it in two steps.

Principle of Addition

 If there are x different ways to do a work and y different ways to two another work and both the works are independent of each other then there are (x+y) ways to do either first OR second work

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Example- 

If we can choose a man in a team by 6 different ways and a woman by 4 different ways then we can choose either a man or a woman by 6+4=10 different ways.

The principle of Multiplication

If there are x different ways to do a work and y different ways to do another work and both the works are independent of each other then there are (x.y) ways to do both first AND second works.

Example

If we can choose a man in a team by 6 different ways and a woman by 4 different ways then we can choose a man and a woman by 6*4=24 different ways.

Definition of Permutation

The process of making different arrangements of objects, letters and words etc by changing their position is known as permutation

Example

A, B, and C are four books then we can arrange them BY 6 DIFFERENT WAYS  ABC, ACB BCA, BAC CAB CBA. so we can say that there are 6 different permutations of this arrangement.

Number of Permutations of n different objects taken all at a time

If we want to arrange n objects at n different places then the total number of ways of doing this or the total number of permutations = {}^n{p_n}

=n! here P represents permutations

Number of Permutations of n different objects taken r at a time

 If we want to arrange n objects at r different places then the total number of ways of doing this or the total number of permutations =  {}^n{p_r}

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=  \frac{{n!}}{{\left( {n - r} \right)!}}    here  {}^n{p_r} represent permutations of n objects taken r at a time.

Number of Permutations of n  objects when all objects are not different

 If we have n objects in total out of which p are of one type, q are of another type, r are of any other type, remaining objects are all different from each other, the total number of ways of arranging them=  \frac{{n!}}{{p!q!r!}}

Number of Permutations of n different objects taken all at a time when repetition of objects is allowed

If we want to arrange n objects at n different places and we are free to repeat objects as many times as we wish, then the total number of ways of doing this or the total number of permutations = {n^n}

Number of Permutations of n different objects taken r at a time when repetition of objects is allowed

 If we want to arrange n objects at r different places(taking r at a time) and we are free to repeat objects as many times as we wish, then the total number of ways of doing this or the total number of permutations= {n^r}

Circular Permutations- When we talk about arrangements of objects, it usually means linear arrangements. But if we wish, we can also arrange objects in a loop. Like we can ask our guests to sit around a round dining table. These types of arrangements are called circular permutations.

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If we want to arrange n objects in a circle, then the total number of ways/circular permutations=(n-1)! this case works when there is some difference between clock-wise and anti-clockwise orders

IF there is no distinction between clock-wise and anti-clockwise orders, the total number of permutations=(n-1)!/2

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Restricted Permutations

There may be following cases of restricted permutation

(a)   Number of arrangements of ‘n’ objects, taken ‘r’ at a time, when a particular object is to be always included     = r{.^{n - 1}}{P_{r - 1}}

(b) Number of arrangements of ‘n’ objects, taken ‘r’ at a time, when a particular object is fixed: =  ^{n - 1}{P_{r - 1}}

(c) The number of arrangements of ‘n’ objects, taken ‘r’ at a time, when a particular object is never taken: = n-1 Pr.

(d) The number of arrangements of ‘n’ objects, taken ‘r’ at a time, when ‘m’ specific objects always come with each-other =  m!{\rm{ }}{\rm{.}}\left( {{\rm{ }}n - m + 1} \right){\rm{ }}!

(e) The number of arrangements of ‘n’ things, taken all at a time, when ‘m’ specific objects always come with each other= \begin{array}{lllllllllllllll} {n{\rm{ }}!{\rm{ }} - {\rm{ }}\left[ {{\rm{ }}m!\left( {n - m + 1} \right)!{\rm{ }}} \right]}\\ \end{array}

In my next post, I will discuss in detail about combinations and will share a large worksheet based on  P & C. In the meantime you can download and solve these questions.

Also, Check the below-given post on Permutation and Combination

 Permutation Worksheet.pdf

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