We had to order 3 people out of 8. To do this, we started with all options 8 then took them away one at a time 7, then 6 until we ran out of medals. Unfortunately, that does too much! And why did we use the number 5? Because it was left over after we picked 3 medals from 8. So, a better way to write this would be:. If we have n items total and want to pick k in a certain order, we get:. And this is the fancy permutation formula: You have n items and want to find the number of ways k items can be ordered:.
Combinations are easy going. You can mix it up and it looks the same. In fact, I can only afford empty tin cans. Well, we have 3 choices for the first person, 2 for the second, and only 1 for the last. Indeed I did. So, if we have 3 tin cans to give away, there are 3! If we want to figure out how many combinations we have, we just create all the permutations and divide by all the redundancies. Writing this out, we get our combination formula , or the number of ways to combine k items from a set of n:.
In other words:. Now we do care about the order. It has to be exactly More generally: choosing r of something that has n different types, the permutations are:. 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. Example: in the lock above, there are 10 numbers to choose from 0,1,2,3,4,5,6,7,8,9 and we choose 3 of them:. So, our first choice has 16 possibilites, and our next choice has 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? Answer: we use the " factorial function ". The factorial function symbol:!
But when we want to select just 3 we don't want to multiply after How do we do that? There is a neat trick: we divide by 13! 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.
The answer is:. Another example: 4 things can be placed in 4! 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 :.
Notice the formula 16! So choosing 3 balls out of 16, or choosing 13 balls out of 16, have the same number of combinations:. Also, knowing that 16! We can also use Pascal's Triangle to find the values. Go down to row "n" the top row is 0 , and then along "r" places and the value there is our answer.
Here is an extract showing row
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