A Complete Guide to Permutations and Combinations

Permutations and Combinations

 Permutations and Combinations, this article will discuss the concept of determining, in addition to the direct calculation, the number of possible outcomes of a particular event or the number of set items, permutations and combinations that are the primary method of calculation in combinatorial analysis.

Common mistakes while learning Permutations and Combinations

There is always confusion amongst the student between permutations and combinations because both are related to the number of the arrangement of different objects and the number of the possible outcome of a particular event or number of ways to get an element from a set. The topic of permutation & combination with examples and the difference between them with justification will be discussed here.

A simple and handy technique to remember the difference between the permutations and combinations is: a permutation is related with the order means the position is important in permutation while the combination is not related with the order means the position is not important in combination.

Before the discussion of permutations and combinations, we require some prerequisites, which are frequently used.

 What is Factorial

          Factorial is the product of the positive integers from 1 to n (counting 1 and n) denoted by n! and read as n factorial is described as below

n!=1.2.3.4……(n-2).(n-1).n=n.(n-1).(n-2)…..3.2.1

^{n}P_{r}=n.(n-1).(n-2)...(n-r+1)=\frac{n!}{(n-r)!}

Mind it 0!=1 

0!=1

1!=1

n!=n(n-l)!

e.g \ \ 3!=3.2.1=6

4!=4.3.2.1=24

5!=5.4!=5.24=120

Counting Methods (Principle of Multiplication and addition)

      Principle of addition: If no two events can happen at the same time, then one of the events can happen in

n1  + n2  + n3  +・ ・ ・.ways

      Principle of Multiplication: Considering that if the events occurred one after the other, then all the events can happen in the order indicated in:

n_{1}.n_{2}.n_{3} …… ways

Example: If an Institute runs 7 different art courses,  3 different technical courses, and 4 different physical courses.

If a student wants to enroll one of each type of course then the number of ways would be

m=7.3.4=84

If a student wants to enroll just one of the courses, then the number of ways would be

n=7 + 3 + 4=14

What is Permutation

The different positioning of the objects are called Permutations, where the order of the arrangement matters. Any positioning of a set of n different objects in a given order is called a permutation of the object.

        Consider an example of the set of letters {P,Q,R,S}, then

  Some of the permutations of the four alphabets taken 4 at a glance are QSRP, SRQP and PRSQ

Any ordering of any r<=n of these particular objects in a specific order is called an “r-permutation” or “a permutation of the n objects taken r at a time.

Basically we like those number of such permutations without set down them.

Example of Permutation Formula

The number of permutations of n different objects taken r at a time will be indicated by

^{n}P_{r}=n.(n-1).(n-2)...(n-r+1)=\frac{n!}{(n-r)!}

In mathematics this is denoted by different ways, some of them are mentioned below:

P(n,r), nPr,Pn,r ,or (n)r

EXAMPLE: Calculate the number m of permutations of six objects, say A, B, C, D, E, F taken three at a glance.

Solution:   Here n=6, r=3, m=?

^{n}P_{r}=\frac{n!}{(n-r)!}

m=^{6}P_{3}=\frac{6!}{(6-3)!}=\frac{6!}{3!}=\frac{3!.4.5.6}{3!}=4.5.6=120

So m=120

EXAMPLE: How many words can be generated by using 2 letters from the word “MATHS”?

Solution: Here n=5, r=2, m=?

^{n}P_{r}=\frac{n!}{(n-r)!}

m=^{5}P_{2}=\frac{5!}{(5-2)!}=\frac{5!}{3!}=\frac{3!.4.5}{3!}=4.5=20

so the required number of words are 20.

What do you understand by a Combination?

A combination for n different elements taken r at a time is any selection of r-th elements where orders are not being considered. Such a selection is called an r-combination. In brief, a Combination is a selection in which the order of the objects selected is not important.

      The Combination gives the number of ways a particular set can be arranged, where the order of the arrangement does not matter.

 To understand the situation of Combination, consider the example

Twenty people arrive in a hall and everyone shakes hand with all the others. How can we get the number of handshakes?  “A” shaking hands with B and B with A will not be two different handshakes. Here, the order of handshake is not important. The number of handshakes will be the combinations of 20 different things taken 2 at a time.

Combination Formula with a simple example

       The number of such combinations will be denoted by

^{n}C_{r}=\frac{n!}{r!(n-r)!} =\frac{^{n}P_{r}}{r!}=\binom{n}{r}

Sometimes it is also denoted by C(n,r), nCr , Cn,r or Crn

Example: A class contains 10 students with 6 men and 4 women. Find the number n of ways to choose a 4-member committee among those students.

This is related to combinations, not permutations, since order is not an important factor in a committee. There are “10 choose 4” such committees. That is:

^{n}C_{r}=\binom{n}{r}=\frac{n!}{r!(n-r)!}

here n=10, r= 4

^{10}C_{4}=\binom{10}{4}=\frac{10!}{4!(10-4)!}=\frac{10.9.8.7.6!}{4.3.2.1.6!}=210

so in 210 ways we can choose such 4-member committee.

Example: A container has 6 blue balls and 8 red balls. Identify the number of ways two balls of any of the colors can be drawn from the container.

Here possibly “14 choose 2” ways for selecting 2 of the 14 balls. Thus:

^{n}C_{r}=\binom{n}{r}=\frac{n!}{r!(n-r)!}

Here n=14 , r=2

^{14}C_{2}=\binom{14}{2}=\frac{14!}{2!(14-2)!} =\frac{14.13.12!}{2.1.12!}=91

so in 91 ways two balls can be drawn of any color.

Difference between Permutation and Combination

The difference between permutation vs combination is briefly given here

PermutationCombination
Order is ImportantOrder is not Important
Order countsOrder does not count
Used for arrangements like electing president, vice president, and treasurerUsed for selection like selecting teams and committee without positions
For electing first, second and third specific positionsFor selecting any three random
For arranging the cards or balls with position and colorFor selecting any color and position
Difference between Permutations and Combinations

Where to apply Permutations and Combinations

  This is the important step that should be kept in mind that whenever the situation is for arrangement, ordering and uniqueness we have to use Permutation and whenever the situation is  for selection, choosing, picking and combination without the concern of order we have to use Combination. If you keep these basics in your mind there will be no confusion “what to use and what not” whenever a question arises.

Use of Permutations and Combinations in real life with examples

In real life permutation and the combination is used in almost everywhere because we know that in real life there would be a situation when order is important and somewhere order is not important, in those situations we have to use the corresponding method.

For example

Find the number N of teams of 11 with a given captain that can be selected from 26 players.

Frequently Asked Questions – FAQs

What is factorial?

The product of the positive integers from  1 to n (including 1 & n )

n!=1.2.3 …….. \left ( n-2 \right ).\left ( n-1 \right ).n

What is a permutation?

The different ordering of the objects are called Permutations

What is a Combination?

     The Combination provides the number of ways a specific set can be set out, where the order of the arrangement does not matter.

Application of permutations and combinations in practical life

A Permutation is used for arrangement or selection of lists where the order is important, and Combination is used for selection or choice where the order is not important.

Permutation formula

^{n}P_{r}=\frac{n!}{(n-r)!}

Combination formula

^{n}C_{r}=\binom{n}{r}=\frac{n!}{r!(n-r)!}

Is there any relation between permutations and Combinations?

Yes,

^{n}C_{r}=\frac{^{n}P_{r}}{r!}

Can we use Permutations and combinations in real life?

Yes,

In the arrangement of words, alphabets, numbers, positions and colours etc. where the order is important permutation will be used

In the selection of committee, teams, menu, and subjects etc where the order is not important combination will be used.

Conclusion

   The brief information about permutations and combinations with basic formula is given read twice or thrice till you get the idea about the concept, in consecutive articles we will discuss in detail the different results and formulae with suitable examples of permutations and combinations. If you want further study go through:

For more Topics on Mathematics, please follow this link.

1.   SCHAUM’S OUTLINE OF Theory and Problems of DISCRETE MATHEMATICS

2.   https://en.wikipedia.org/wiki/Permutation

3.   https://en.wikipedia.org/wiki/Combination

4.   https://in.bgu.ac.il/

5.   https://www.cs.bgu.ac.il/

About DR. MOHAMMED MAZHAR UL HAQUE

I am DR. Mohammed Mazhar Ul Haque , Assistant professor in Mathematics. Having 12 years of experience in teaching. Having vast knowledge in Pure Mathematics , precisely on Algebra. Having the immense ability of problem designing and solving. Capable of Motivating candidates to enhance their performance.
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