permutation and combination?

Question

In your own words, explain the difference between a permutation and a combination. Also, give an example of a problem whose solution is a permutation and an example of a problem whose solution is a combination.

Answer 1

Permutation means the arrangement and order of things, and is used if the order is considered. Combination refers to a slection of things in which the order of things doesn't matter.

For example, say we have to create a 3-digit number using the digits "4, 5, 6, 7". To form this number, the digits must be arranged. Different numbers will be formed depending on the order that we arrange the digits into. This is a permuation.

For a combination, say that we have to make a team of 11 players out of 20 players. It's a combination because the order of the players in the team doesn't matter. No matter what order we list the players, the team will stay the same.

A permutation is written like this:
nPr = n!/(n-r)!
Number of permutations of (n) different things taken (r) at a time.

For example, how many different signals can be made by 4 flags from 7-flags of different colours? Number of ways taking 4 flags out of 7-flags = 7P4.

7P4 = 7!(7-4)!
7 x 6 x 5 x 4 = 840

A combination is written like this:
nCr
3C2 = 3 x 2/2!
8C3 = 8 x 7 x 6/3!
You put 'n' into a factorial like normal, but you only multiply 'r' number of digits down. For example, 5C3. 5! = 5 x 4 x 3 x 2 x 1. But since 'r' equals 3, you only go down the three numbers to make 5 x 4 x 3.

For example, how many different teams of 4 can you make out of 10 people? The number of ways of making teams of 4 with 10 people = 10C4

10C4 = 10 x 9 x 8 x 7/4!
10C4 = 10 x 9 x 8 x 7/4 x 3 x 2 x 1
10C4 = 210

Hope this helps! This took a long time to write. :P

Answer 2

As has been suggested, the main difference is whether the order in which things are selected matters. If the order is important, then use a permutation--if the order is unimportant, use a combination.

Here is my example. Suppose that you have 10 people who are members of a club. They need to select two people to serve as officers--a president and a vice president. How many ways can this be done? 90--which we get from using the permutation 10 x 9. Why a permutation? Because electing Betsy president and Frank vice president is a very different outcome than electing Frank president and Betsy vice president.

But suppose we need two of these ten people to go save everyone seats in the movie theater. How many ways can two people be selected for this task? 45--using the combination 10 x 9 / 2. Why a combination? Because choosing Betsy first and Frank second yields the same result as choosing Frank first and Betsy second. What is important in this case is the COMBINATION of people who end up being selected.

Does that help?

Answer 3

In a permutation, order matters. In a combination, order does not matter. The number of ways you can seat 5 people in 4 chairs in a row is a permutation since order matters. P(5,4)=5x4x3x2=120
The number of different teams of 4 players that can be chosen from 7 people is a combination since it doen't matter who is chosen first or last. C(7,4)=7x6x5x4
I hope this helps explain the difference.

Answer 4

This site is a good reference on permutations and combinations:
http://www.themathpage.com/aPreCalc/permutations-combinations.htm
It also has some solved problems that illustrate the difference between the two.
This article also helps explain the difference:
http://en.wikipedia.org/wiki/Permutations_and_combinations

So all you need to do now is put the definition in your own words.

Answer 5

Permution is like combinations only that allows any order of the numbers and it count as different sets.

For ex: 0123, 0132, 0231, 0213, etc...

However Combinations doesn't allow this

0123 and 0132 is counted as the same thing, but 0123 and 0124 is a different combination.

10 numbers and 4 ways to put them

Permutation = 5040

Combination = 210

Permutation Formula
nPr = (n!)/((n - k)!)

Combination Formula
nCr = (n!)/(n!(n - k)!)

So the difference between them is the (1/(n!))