Combinations and Permutations

We use the word combination mostly during our interaction to say things like, “my fruit mix juice is a combination of apple, bananas and guava”. In this case, the order of the fruits doesn’t matter.

But when you say the combination to the password is “3987”, the order of the numbers does matter here. In other words, you can even say that permutation is nothing but an ordered combination. Hence, if the order matters, it is called permutation and if the order doesn’t count, then it is called as combination. Hope you have now got the basic idea behind combination and permutation?

Combinations

To calculate the number of combinations of n objects taken r at a time, is identified using the formula:

C(n,r)=n! / (n−r)! r!

Say, for example, if you need a pick a team of only 3 people out of a group of 10, then the combination formula will work out like:

C (10,3) = 10! / (10-3)! 3!
C (10,3) = 10! / 7! x 3!
C (10,3) = 10x 9 x 8 / 3 x 2 x1
C (10,3) = 720 / 6
C (10,3) = 120

So, if you need to choose a team of 3 (r) people out of a group of 10 (n), then you can have a total of 120 combinations, if you don’t care about the order in which they are selected.

Permutations

To calculate the number of permutations of n objects taken r at a time, is identified using the formula:

P(n,r)=n! / (n−r)!

Now if we want to get the permutation for choosing a team of 3 people from a group of 10, let’s take a look at the permutations formula and see how that works out.

P (10,3) = 10! / (10-3)!
P (10,3) = 10! / (7)!
P (10,3) = 10. 9. 8. 7. 6. 5. 4. 3. 2. 1 / 7. 6. 5. 4. 3. 2. 1
P (10,3) = 10. 9. 8
P (10,3) = 720

Examples

Now let’s take a look at some more examples:

Examples 1: How many different signals can be made by 3 flags from 5 flags of different colors?

So, let’s use our formula to get the permutation and combination:

P(5,3) = 5! / (5-3)!
P(5,3) = 5. 4. 3. 2. 1 / 2. 1
P(5,3) = 5. 4. 3
P(5,3) = 60

So, we can make 60 different signals made by 3 flags from 5 flags of different colors.

Examples 2: How many words can you generate using the letters from the word ‘SIMPLETON’?

There is a total of 9 different letters in the word ‘SIMPLETON’. Let’s put this in the formula:

P (9,9) = 9! / (9-9)!
P (9,9) = 9. 8. 7. 6. 5. 4. 3. 2. 1
P (9,9) = 362880

The total number of permutations to generate different words from the word ‘SIMPLETON’, taking all the letters at a time, is 362880.

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