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When Order Is Not Relevant
Combinations problems are very similar to permutations problems. The key distinction
is that placement/order isn’t relevant for combinations, but placement/order
is relevant for permutations.
For example, if a committee is being put together for a company and there is
a president, vice president, and treasurer, order matters. If that same committee of three people
is being put together but nobody has a rank then order does not matter because
there is no hierarchy.
To illustrate this, we will show a permutation question and then make a slight change to make it a combinations question.
I. Example of Permutation vs. Combination
Let’s say we play a game of dice where we roll two dice (one red and one blue) and record the results. If the dice roll up the same number, the results aren’t counted and we roll again. The result is that there are no doubles. How many permutations are possible?
1. Figure out how many places there are to fill
Because there are two dice, there are two places to fill: __ __
2. Figure out how many objects potentially can go into each place
Since you cannot get the same result (doubles) on the second die there are only 5 possibilities for the second roll.
_6_ _5_
3. Multiply for the answer
_6_ × _5_ = 30
| (Red,Blue) |
(Red,Blue) |
(Red,Blue) |
(Red,Blue) |
(Red,Blue) |
(Red,Blue) |
(1,1) |
(2,1) |
(3,1) |
(4,1) |
(5,1) |
(6,1) |
(1,2) |
(2,2) |
(3,2) |
(4,2) |
(5,2) |
(6,2) |
(1,3) |
(2,3) |
(3,3) |
(4,3) |
(5,3) |
(6,3) |
(1,4) |
(2,4) |
(3,4) |
(4,4) |
(5,4) |
(6,4) |
(1,5) |
(2,5) |
(3,5) |
(4,5) |
(5,5) |
(6,5) |
(1,6) |
(2,6) |
(3,6) |
(4,6) |
(5,6) |
(6,6) |
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