Question 134563
The question is "does order matter?"


For example:


Let's say I have a box of 8 differently colored crayons.  How many ways can I select 3 crayons from the box of 8?  The answer depends on whether order matters or not, that is to say is 'blue, green, orange' a different selection than 'green, orange, blue'?  And the answer to that depends on the actual situation you are trying to enumerate.


Here is another example:  Let's say that you just started a Probability Calculators Club at school.  You have 10 members and 4 club officer positions.  In keeping with the theme of the club, you decide to select the officers at random instead of holding elections.  You put slips of paper with each member's name into a bag, and the first one drawn becomes the president, the second becomes vice president, and so on.


In this case, order definitely matters, because 'Suzy, Billy, Tommy, Ann' is a different set of officers than 'Tommy, Suzy, Ann, Billy'


In this situation, you would use a permutation,


P(n,r) = {{{n!/(n-r)!}}}.


In the crayon box example, it probably doesn't matter which color comes out of the box first, second, or third.  The selection of 'blue, green, orange' would most likely be considered the same as 'green, orange, blue'  In this case you would use a combination:


C(n,r) = {{{n!/(r!(n-r)!)}}}


The difference between the two is that factor of {{{r!}}} in the denominator of the combination formula.  {{{r!}}} is the number of ways any given selection can be ordered, and represents the number of times things that are the same except for order are over-counted when order doesn't matter.  In the crayon box example, there are {{{3!=6}}} ways that you can put 'green, orange, and blue' in different orders, so if you just use the permutation formula, you will count those three colors as 6 different selections, even though we decided that order doesn't matter.  So you have to divide the number developed in the permutation formula by the number of possible orders of the selection of r objects, hence the {{{r!}}} in the denominator of the combination formula.