Permutation and Combination

What's the Difference?

I’ve always confused “permutation” and “combination” — which one’s which?

Permutation:  

Permutation means arrangement of things. The word arrangement is used, if the order of things is considered.

Combination:  

Combination means selection of things. The word selection is used, when the order of things has no importance.

Example:     

Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4, To form this number the digits have to be arranged. Different numbers will get formed depending upon the order in which we arrange the digits. This is an example of Permutation.

Now suppose that we have to make a team of 11 players out of 20 players, this is an example of combination, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed.

ALWAYS REMEMBER!

It is very important to make the distinction between permutations and combinations. In permutations, order matters and in combinations order does not matter. The important information can be summarized by:

Arranging Objects

The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). 

n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

Example:

How many different ways can the letters P, Q, R, S be arranged?
The answer is 4! = 24.

This is because there are four spaces to be filled: _, _, _, _
The first space can be filled by any one of the four letters. 
The second space can be filled by any of the remaining 3 letters. 
The third space can be filled by any of the 2 remaining letters.
The final space must be filled by the one remaining letter. 

The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!

• The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is:

   n!        

p! q! r

Combination

The number of ways of selecting r objects from n unlike objects is:

Example:

There are 10 balls in a bag numbered from 1 to 10. 
Three balls are selected at random. 
How many different ways are there of selecting the three balls?

¹⁰C₃ =10!=10 × 9 × 8= 120
             3! (10 – 3)!3 × 2 × 1

Permutation

A permutation is an ordered arrangement. 

The number of ordered arrangements of r objects taken from n unlike objects is:

ⁿP_r =       n!      
          (n – r)!

Example:

In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.

¹⁰P₃ =10!
            7!

= 720

There are therefore 720 different ways of picking the top three goals.


Probability
The above facts can be used to help solve problems in probability.

Example:

In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?

The number of ways of choosing 6 numbers from 49 is ⁴⁹C₆ = 13 983 816 .
Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf)

which is about a 1 in 14 million chance.

 

Now, Let's try some exercises!


References:
http://tutors4you.com/permutationcombinationtutorial.htm 
http://revisionmaths.com/advanced-level-maths-revision/statistics/permutations-and-combinations