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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 choice, 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 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? The factorial function symbol: But when we want to select just 3 we don't want combination formula with repetition multiply after How do we do that? There is a neat trick: This is how lotteries work. The numbers are drawn one combination formula with repetition 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 combination formula with repetition 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 values.

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, combination formula with repetition 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 flavors, 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 combination formula with repetition 1st to 5th container. In other words it is now like the pool balls question, but with slightly changed combination formula with repetition. 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 combination formula with repetition 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, we 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 that multiplying no numbers together gets us 1, but it helps simplify a lot of equations.

Example Our combination formula with repetition 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 combination formula with repetition. These are the possibilites: It is often called "n choose combination formula with repetition such as "16 choose 3" And is also known as the Binomial Coefficient. Pool Balls without order So, our pool ball example now without order is:

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First give everybody three pens, using 75 of them. Selection with Repetition The permutation and combination question we have done so far are basically about selecting objects. Something in particular that was missing: How many distinct values can be represented with 5 decimal digits? Here we are selecting items digits where repetition is allowed: We can actually answer this with just the product rule: Same as other combinations: Same as permutations with repetition: You walk into a candy store and have enough money for 6 pieces of candy.

The store has chocolate C , gummies G , and horrible Chinese candy H. How many different selections can you make? Here are some possible selections you might make: C then G then H. We don't want our candy to mix: How many solutions does this equation have in the non-negative integers? We select of them with repetition and that gives us a solution to the equation. The basic approach is: How do we deal with the repeated P? Let's pretend they're different: Then in the remaining 3 positions, permute the remaining 3 elements.

How many ways for a game that is played with two decks shuffled together? There are 11 letters, but four Is, four Ss, two Ps. More Counting Examples Your mother-in-law buys small gifts to give to relatives for Christmas, for reasons you don't understand. Each of the things is different, because she spends too much time shopping. There are 25 relatives to give gifts to.

How many ways are there to distribute the gifts? There's no fairness in this family In the above scenario, how many ways can the gifts be distributed so each person gets 40 items?

The next year, you are sent to the basement to arrange the gifts into 25 piles, with one for each person. You don't know who will be getting each pile, so the order of the piles doesn't matter.

How many arrangements are there for you to choose from? The next year, your mother-in-law buys identical pens to give to the family. You start to wonder what is going on. How many ways can they be distributed to 25 family members? How many ways if each person must get at least 3 pens? How many ways if each person gets 40 pens?

Solutions This is a permutation with repetition. For each gift, it can be given to one of 25 family members. First gift in 25 ways, second gift in 25 ways, …. Think of giving each family member 40 tokens with their name on it. Line the different gifts up in a row and have each family member drop their 40 tokens on gifts.

To distribute the gifts, we must select people to get each one. Order doesn't matter here, since the pens are identical. This is a combination with repetition problem: How many of those are there? We should first decide on a way to order permutations. Then we can generate them in that order, and be sure we found them all.

We will order them based on lexicographic order. Two permutations are equal if all elements are the same. The following algorithm will generate all permutations of elements of a set, in lexicographic order: Easy enough to prove by induction. To generate all combinations of a set we make this observation: The following algorithm will generate all combinations of elements of a set: An example of recursion making things easier: Which is more clear?

Which one do you believe is correct? I have had to do that zero times in my life. I have had to generate all of them.