Permutation and Combination

What are Permutations and Combinations?

Permutations are a set of ordered objects. The word “combinations” has slipped into English usage for things like a “combination lock”. The kind of lock you put around your bicycle should be called a “permutation lock,” because the order does matter.

If you don’t care what order you have things, it’s a combination. If you do care, it’s a permutation. Lottery tickets where you pick a few numbers are a combination. That’s because the order doesn’t matter (but the numbers you select do). Picking winners for a first, second and third place raffle is a permutation, because the order matters.

What is the Permutation Formula?

 
In general P(n, r) means that the number of permutations of n things taken r at a time. We can either use reasoning to solve these types of permutation problems or we can use the permutation formula.

The formula for permutation is

 

Permutations Word Problems 

Example 1:

In how many ways can the letters of the word APPLE can be rearranged?

Solution:

Total number of alphabets in APPLE = 5.

Number of repeated alphabets = 2

Number of ways APPLE can be rearranged = 5!2! = 60.

The word APPLE can be rearranged in 60 ways.

Example 2:
Ellie want to change her password which is ELIIE9 but with same letters and number. In how many ways she can do that?

Solution:

Total number of letters = 6.

Repeated letters = 2 Is and 2 Es.

Number of times ELLIE9 can be rearranged = 6!2!2! = 6×5×3×2×1 = 180.

But the password need to be changed. So, the number of ways new password can be made = 180−1=179

Example:

In the match of the Day's goal of the month competition, you had to pick the top 3 goals out of 10.

¹⁰P₃ = 10!/7!

         = 720

There are therefore 720 different ways of picking the top three goalss

COMBINATION

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination. Combination doesn't have any rules. A combination is a way of selecting items from a collection, such that (unlike permutation) the order of selection does not matter. In smaller cases it is possible to count the number of combinations.

Combination Formula
A formula for the number of possible combinations of r objects from a set of n objects. This is written in any of the ways shown below.

 

All forms are read aloud "n choose r."

Formula:

	Note: , where nPr is the formula for permutations of n objects taken r at a time.
Example:	How many different committees of 4 students can be chosen from a group of 15?
Answer:	There are possible combinations of 4 students from a set of 15.
	There are 1365 different committees.

	Question!
	In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?
	From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. In how many ways can it be done?
	There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects such that each subject is allowed at least one period?