Show In English we use the word "combination" loosely, without thinking if the
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**.
In other words: A Permutation is an
## PermutationsThere are basically two types of permutation: **Repetition is Allowed**: such as the lock above. It could be "333".**No Repetition**: for example the first three people in a running race. You can't be first and second.
## 1. Permutations with RepetitionThese are the easiest to calculate. When a thing has For example: choosing
More generally: choosing
(In other words, there are Which is easier to write down using an exponent of
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, the formula is simply:
## 2. Permutations without RepetitionIn this case, we have to
## 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, 12, 11, ... etc. And the total permutations are:
But maybe we don't want to choose them all,
In other words, there are 3,360 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"
So, when we want to select
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
That was neat: the The formula is written:
16! (which is just the same as: 10! (which is just the same as: ## NotationInstead of writing the whole formula, people use different notations such as these: P(n,r) = nPr = nPr = n! ## Examples:- P(10,2) = 90
- 10P2 = 90
- 10P2 = 90
There are also two types of combinations (remember the order does **Repetition is Allowed**: such as coins in your pocket (5,5,5,10,10)**No Repetition**: such as lottery numbers (2,14,15,27,30,33)
## 1. Combinations with RepetitionActually, these are the hardest to explain, so we will come back to this later. ## 2. Combinations without RepetitionThis 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:
So, the permutations 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:
(Another example: 4 things can be placed in So we adjust our permutations formula to n! That formula is so important it is often just written in big parentheses like this:
It is often called "n choose r" (such as "16 choose 3") And is also known as the Binomial Coefficient. ## NotationAll these notations mean "n choose r": C(n,r) = nCr = nCr = (n Just remember the formula: n!
So, our pool ball example (now without order) is: 16! = 16! = 20,922,789,888,000 = 560 Notice the formula 16! So choosing 3 balls out of 16, or choosing 13 balls out of 16, have the same number of combinations: 16! In fact the formula is nice and n! Also, knowing that 16!/13! reduces to 16×15×14, we can save lots of calculation by doing it this way: 16×15×14 = 3360 = 560 ## Pascal's TriangleWe 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 16: 1 14 91 364 ... 1 15 105 455 1365 ...
1 16 120 ## 1. Combinations with RepetitionOK, now we can tackle this one ... Let us say there are five flavors of icecream: We can have three scoops. How many variations will there be? Let's use letters for the flavors: {b, c, l, s, v}. Example selections include - {c, c, c} (3 scoops of chocolate)
- {b, l, v} (one each of banana, lemon and vanilla)
- {b, v, v} (one of banana, two of vanilla)
(And just to be clear: There are Now, I can't describe directly to you how to calculate this, but I can show you a 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 meansmove, circle means scoop). In fact the three examples above can be written like this:
So instead of worrying about different flavors, we have a 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). So (being general here) there are This is like saying "we have
Interestingly, we can look at the arrows instead of the circles, and say "we have (r + n − 1)! So, what about our example, what is the answer? (3+5−1)! There are 35 ways of having 3 scoops from five flavors of icecream. ## In ConclusionPhew, that was a lot to absorb, so maybe you could read it again to be sure! But knowing how these formulas work is only half the battle. Figuring out how to interpret a real world situation can be quite hard. But at least you now know the 4 variations of "Order does/does not matter" and "Repeats are/are not allowed":
708, 1482, 709, 1483, 747, 1484, 748, 749, 1485, 750 Copyright © 2021 MathsIsFun.com |