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:
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