Permutations and Combinations

Permutations:

A permutation is a grouping of items in which ORDER DOES MATTER.  For example, if we are asked how many permutations we can make with the letters A, B, and C, A-C-B is different from C-B-A.  So how many permutations can we make?  

A-B-C

A-C-B

B-A-C

B-C-A

C-A-B

C-B-A

The answer is 6.  But what if we were asked to find the number of permutations of the first 15 letters of the alphabet?  Clearly writing all these out would be difficult.  So we need a formula, and luckily we have one:

P = n!

So the total number of permutations for the first 15 letters of the alphabet is 15! = 15*14*13*12* ..... 2*1 =  .  Good thing we didn't try to write them out.  

Now suppose you are asked to find the number of permutations that you can create from the first 9 letters of the alphabet, while only using 5 letters at a time.  In other words, how many 5-letter words can you create from the first 9 letters of the alphabet?  We need to modify our original formula to get:

P =

where n is the total number of items to chose from (9) and r is the total number taken for each group (5).  Remember that you cannot have the factorial of a negative number, so if you forget which is n and which is r, r must be the smaller of the two.  

So for our example, we have:

P =  =  = 15120

You can use a graphing calculator for this as well.  In a TI-82, enter 9, then press Math, go to the PRB menu, select the item nPr, press 5, and enter.  

Combinations:

A combination is a grouping of items in which ORDER DOES NOT MATTER.  The number of combinations that can be made out of the letters A, B, and C, using all 3 letters, is only 1.  Since the order does not matter A-B-C is the same as C-A-D.  The formula for combinations is:

C =

Where n is the total number of items to chose from, and r is size of the group.  You could also use the nCr function of your calculator (in the MATH -> PRB menu in most TI's).   

Example:

Suppose you have 25 cd's and you want to know how many different groups you can select for your 6-disc changer.  Since you only listen to them on shuffle, the order you put them into your changer does not matter.  How many groups can you make?

Answer:

Since order does not matter, you would use the equation for combinations, where n = 25 and r = 6.  

C =   = 177,100.

That's a lot of combinations!

That's all there is to it.  Permutations are groupings where order DOES matter, combinations are those where order does NOT matter.  Just use the appropriate formulas.  If you make a mistake and switch n and r, your calculator will give you an error since you'll be asking it for the factorial of a negative number.  If you get an answer, as long as you use the right formula, chances are it's right.