what is the difference between a combination and a permutation. A simple answer please?

Answer 1

Both permutations and combinations are methods of ordering a group of things, but they differ in that:

Permutation = order matters
Combination = order does not matter

Let me illustrate a simple example that shows both scenarios:

6 people, Adam, Bob, Chris, Don, Eric, and Frank, ran a race. A medal is awarded for 1st (gold) 2nd (silver) and 3rd (bronze). Determine how many ways the 6 runners can receive EACH medal.

As you can see, order does matter in this scenario. To solve for the permutation, use the following formula:

P(n,k) = n!/(n-k)! where:
n = # of choices
k = # of things to place

In this case, n = 6 and k =3, thus P(6,3) = 6!/(6-3)! = 6*5*4 = 120

For a combination, order does not matter. Let me use the same example to illustrate:

6 people, Adam, Bob, Chris, Don, Eric, and Frank, ran a race. A medal is awarded for 1st (gold) 2nd (silver) and 3rd (bronze). Determine how many ways the 6 runners can receive ANY medal.

Let's say Adam, Bob, and Chris were the first 3 runners that finished and they finished with the following medals:

Adam = gold
Bob = silver
Chris = bronze

Since the problem states how many ways the runners can win ANY medal, the following scenario would be the same as the above one:

Adam = silver
Bob = bronze
Chris = gold

All that matters is that Adam, Bob, and Chris one a medal, thus the 2nd scenario would not matter to the # of combinations. If this was a permutation on the other hand, then it would count. The formula for solving combinations is:

C(n,k) = n!/[(n-k)!k!]

Using n = 6, k = 3, you have:

C(6,3) = 6!/[(6-3)!3!] = 6*5*4/(3*2) = 20 ways

The k! term in the combination eliminates the repetition of groups with different order, like the example mentioned above.

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Hope this helps

Answer 2

A permutation is the way to can arrange things:

ex:

abc
bca
cab
cba
acb
bac

are the ways to arrange 3 things: 3P3 = 3! = 3*2*1 = 6

There are n! ways to arrange n things

What if I chose 3 things out of 5 and arrange them, which is the same as saying that order matters:

5P3 = 5!/(5-3)! = 5*4*3 = 60 so there are 60 ways to choose 3 letters from the letters a,b,c,d,e and arrange them

Now, for combinations. Lets say that I don't care how they are arranged

Then 5C3 = 5P3/3! = 60/6 = 10 so there are 10 ways to sample 3 from 5 if I don't care about order. Note something interesting: if I choose 3 from 5 then the other 2 are determined. So, there are 5 ways to for the first one left and 4 ways for the second one and then the reverse order : 5*4/2 = 10.

There is symmetry:

nCk = nC(n-k) this shows up in Pascal's triangle.

Pascal's triangle is nCk where n is the row, and k is the column
n and k both start at 0. This symmetry shows up in the triangle.

Nifty, no?

A combination lock should really be called a permutation lock. It does matter what order you put the numbers in!

Answer 3

1) How many different ways 52 cards can be ordered in a deck? That's permutation.

2) How many different 5-card poker hands can be dealt from a deck? That's combination.

With combinations, ordering is not relevant.

Answer 4

Permutation

Arrangement number or order. The idea that distinguishableobjects jay be arrange in various orders

Combination

is a un-ordered collections of unique elements. The order of the elements in a cobination is not important.

Click on the URL below for additional information concerning Combination.

en.wikipedia.org/wiki/Combination

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