Easy Permutations and Combinations
4 stars based on
In English we use the word "combination" loosely, without thinking if the order of things is important. Now we do care about the order. It has to be exactly In other words, there are n possibilities for the first combination and permutation examples, THEN there are n possibilites for the second choice, and so on, multplying each time.
So, our first choice has 16 possibilites, and our next choice combination and permutation examples 15 possibilities, then 14, 13, 12, 11, And the total permutations are:.
In other words, there are 3, different ways that 3 pool balls could be arranged out of 16 balls. But how do we write that mathematically?
The factorial function symbol: But when we want to select just 3 we combination and permutation examples want to multiply after How do we do that? There is a neat trick: 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!
Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order. 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.
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:.
In other words choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations. We can also use Pascal's Triangle to find the combination and permutation examples. Go down to row "n" the top row is 0and then along "r" places and the value there is our answer. Here is an extract showing row Let us say there are five flavors of icecream: And just to be clear: Order does not matter, and we can repeat!
Now, I can't describe directly to you how to calculate this, but I can show you a special technique that lets you work it out.
Think about the ice cream being in boxes, we could say "move past the first box, then take 3 scoops, then move along 3 more boxes to the end" and we will have 3 scoops of chocolate! So it is like we are ordering a robot to get our ice cream, but it doesn't change anything, we still get what we want.
We can write this down as arrow means movecircle means scoop. OK, so instead of worrying about different combination and permutation examples, we have a simpler question: Notice that there are always 3 circles 3 scoops of ice cream and 4 arrows we need to move 4 times to go from the 1st to 5th container.
In other words it is now like the pool balls question, but with slightly changed numbers. And we can write it like this:. But knowing how these formulas work is only half the battle. Figuring out how to interpret a real world situation can be quite hard. Hide Ads About Ads. Combinations and Permutations What's the Difference? So, in Mathematics we use more precise language: When the order doesn't matter, it is a Combination.
When the order does matter it is a Permutation. So, combination and permutation examples should really call this a "Permutation Lock"! A Permutation is an ordered Combination. To help you to remember, think " P ermutation After choosing, say, number "14" we can't choose it again.
And the total permutations are: It may seem funny combination and permutation examples multiplying no numbers together gets us 1, but it helps simplify a lot of equations. Example Our "order of 3 out of 16 pool balls example" is: How many ways can first and second place be awarded to 10 people? For example, let us say balls 1, 2 and 3 are chosen.
These are the possibilites: It is often called "n choose r" such as "16 choose 3" And is also known as the Binomial Coefficient. Pool Balls without order So, our pool ball combination and permutation examples now without order is: