Monday, March 14, 2016

1. Permutations with Repetition

These are the easiest to calculate.
When we have n things to choose from ... we have n choices each time!
When choosing r of them, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)
Which is easier to write down using an exponent of r:
n × n × ... (r times) = nr
Example: in the lock above, there are 10 numbers to choose from (0,1,...9) and we choose 3 of them:
10 × 10 × ... (3 times) = 103 = 1,000 permutations
So, the formula is simply:
nr
where n is the number of things to choose from, and we choose r of them
(Repetition allowed, order matters)

2. Permutations without Repetition

In this case, we have to reduce the number of available choices each time.
For example, what order could 16 pool balls be in?
After choosing, say, number "14" we can't choose it again.
So, our first choice has 16 possibilites, and our next choice has 15 possibilities, then 14, 13, etc. And the total permutations are:
16 × 15 × 14 × 13 × ... = 20,922,789,888,000
But maybe we don't want to choose them all, just 3 of them, so that is only:
16 × 15 × 14 = 3,360
In other words, there are 3,360 different ways that 3 pool balls could be arranged out of 16 balls.
Without repetition our choices get reduced each time.
But how do we write that mathematically? Answer: we use the "factorial function"
The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples:
  • 4! = 4 × 3 × 2 × 1 = 24
  • 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
  • 1! = 1
Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.
So, when we want to select all of the billiard balls the permutations are:
16! = 20,922,789,888,000
But when we want to select just 3 we don't want to multiply after 14. How do we do that? There is a neat trick ... we divide by 13! ...
16 × 15 × 14 × 13 × 12 ...
  = 16 × 15 × 14 = 3,360
13 × 12 ...
 
Do you see? 16! / 13! = 16 × 15 × 14
The formula is written:
where n is the number of things to choose from, and we choose r of them
(No repetition, order matters)

Examples:

Our "order of 3 out of 16 pool balls example" is:
16!  =  16!  =  20,922,789,888,000  = 3,360
(16-3)! 13! 6,227,020,800
(which is just the same as: 16 × 15 × 14 = 3,360)
How many ways can first and second place be awarded to 10 people?
10!  =  10!  =  3,628,800  = 90
(10-2)! 8! 40,320
(which is just the same as: 10 × 9 = 90)

Notation

Instead of writing the whole formula, people use different notations such as these:
Example: P(10,2) = 90

Combinations

There are also two types of combinations (remember the order does not matter now):
  1. Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)
  2. No Repetition: such as lottery numbers (2,14,15,27,30,33)

1. Combinations with Repetition

Actually, these are the hardest to explain, so we will come back to this later.

2. Combinations without Repetition

This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!
The easiest way to explain it is to:
  • assume that the order does matter (ie permutations),
  • then alter it so the order does not matter.
Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order.
We already know that 3 out of 16 gave us 3,360 permutations.
But many of those are the same to us now, because we don't care what order!
For example, let us say balls 1, 2 and 3 are chosen. These are the possibilites:
Order does matter Order doesn't matter
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
1 2 3
So, the permutations will have 6 times as many possibilites.
In fact there is an easy way to work out how many ways "1 2 3" could be placed in order, and we have already talked about it. The answer is:
3! = 3 × 2 × 1 = 6
(Another example: 4 things can be placed in 4! = 4 × 3 × 2 × 1 = 24 different ways, try it for yourself!)
So we adjust our permutations formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more):
That formula is so important it is often just written in big parentheses like this:
where n is the number of things to choose from, and we choose r of them
(No repetition, order doesn't matter

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