Combinations

Combinations are groupings of things where the order of the things does not matter. Remember that with permutations order is important. A chocolate, vanilla and strawberry ice cream cone is different from strawberry, vanilla and chocolate. It matters what flavor is on top! With combinations, life is simpler. Order does not matter. Trail mix combines raisins, peanuts and M&Ms in no particular order.

Here's an example with a group of 5 kids:
DAVID   BRYAN   TONYA   BJ   CARRIE

We want to form 2-person groups to ride a ferris wheel. How many 2-person groups can we make? Well, here they are:

   <1>     <2>     <3>     <4>     <5>
  DAVID   DAVID   DAVID   DAVID   BRYAN
  BRYAN   TONYA     BJ    CARRIE  TONYA

   <6>     <7>     <8>     <9>     <10>
  BRYAN   BRYAN   TONYA   TONYA     BJ
    BJ    CARRIE    BJ    CARRIE  CARRIE
Why are there only 10 groups, when the number of permutations of 2 things out of a set of 5 would be:
5 x 4 = 20?
The reason is that half the permutations (10 of them) are missing! The missing 10 permutations are:

   <1>     <2>     <3>     <4>     <5>
  BRYAN   TONYA     BJ    CARRIE  TONYA
  DAVID   DAVID   DAVID   DAVID   BRYAN

   <6>     <7>     <8>     <9>     <10>
    BJ    CARRIE    BJ    CARRIE  CARRIE
  BRYAN   BRYAN   TONYA   TONYA     BJ
These are the same as the first 10, but the names are switched.
These don't count, because a 2-person group of BRYAN and DAVID is the same as DAVID and BRYAN.

So, the number of combinations of 5 kids riding a ferris wheel 2 at a time is: