Permutations
Last updated: 20 Mar 2009
Permutations are very important in Algebra. They are most often used for finding probabilities or the number of ways something can be arranged. In fact, a permutation is simply the number of different ways a set can be arranged.
An example of a permutation
One of the most used examples of a permutation is the number of ways to arrange a batting order for a baseball team. A baseball team has 9 players, each of which go up and bat before repeating the order.
How many different ways can you organize 9 players in an order? For the first position, there would be 9 options, for the second position 8 options, for the third 7, and so on, until for the 9th position there would be only one option (since no player can go twice before all of the players have gone). Hence, the number of positions would be
From that, we know that there are 362,880 ways to order a batting order for a baseball team. Writing can be very tedious, though. Hence, we have what we call a "factorial"
Factorials
Whenever you multiply a number by the number right before it, then by the number right before that and so on until you multiply it by one, we simply write it as "n!" where n is the number. So 9 factorial, like in the example above, would be written 9!
Permutations where the entire set isn't used
Most of the time, unlike in the baseball example, you will be selecting a few objects out of the set, not every object. The number of objects you take is called r. The size of the set is n. It is expressed as P(n,r), so for example choosing the 1st second and third places out of a field of 9 players would be expressed as P(9,3), since there are 9 objects and three are being selected.
To solve these, you use the formula
So for the top three out of nine, it would be
If you expand the factorials, you get this:
Since they are all being multiplied, you can simply cancel out the same numbers on the top and bottom. Hence, you may remove the 6, 5, 4, 3, 2 and 1 from both top and bottom.
So there are 504 ways to choose 1st 2nd and 3rd place out of 9 players.
Wait, that seems harder than it should be...
It is. Really, you can skip the whole and just multiply the first r objects. so in the case you could have skipped straight to
. It's much simpler. Even so, you should remember the formula for tests.
What to remember:
- A permutation is the number of ways objects can be ordered in a set. Order DOES matter.
- When the whole set is used, it is simply
means
factorial, or
down to 1.
- When you only choose some of the set, simply take the number you are supposed to choose and use that many numbers from the start of the factorial. So
and
- Remember the formula if it helps you .
