Why do combination and permutation have different rules for order?

• MHB
In summary, the conversation has involved 5 finalists, one of which was awarded "Best in Show" and another finalist was awarded "Honorable Mention." There are a total of 20 ways that these awards could be given out.
At a recent dog show, there were 5 finalists. One of the finalists was awarded "Best in Show" and another finalist was awarded "Honorable Mention." In how many different ways could the two awards be given out?

The words "how many ways" reminds of probability. I just don't recall if this is a combination or a permutation.

I will take a guess and say this is a combination problem.

The set up is 5C2.

You say?

Beer soaked ramblings follow.
At a ...
I will take a guess and say this is a combination problem.

The set up is 5C2.

You say?
I say stop guessing.
You have a textbook.

Personally, I would not worry about what it is called! There are 5 dogs anyone of which could be declared "best in show". Once that had been chosen, there are 4 dogs left that could be chosen "honorable mention".

There are a total of 5(4)= 20 ways that could be done.

(Since order, which dog is "best in show" and which is "honorable mention", is relevant, this is a "permutation" problem but, as I say, that is really not important.)

Country Boy said:
Personally, I would not worry about what it is called! There are 5 dogs anyone of which could be declared "best in show". Once that had been chosen, there are 4 dogs left that could be chosen "honorable mention".

There are a total of 5(4)= 20 ways that could be done.

(Since order, which dog is "best in show" and which is "honorable mention", is relevant, this is a "permutation" problem but, as I say, that is really not important.)

Are you saying that that 5C2 should be 5P2?

Let me see.

5P2 = 5!/(5 - 2)!

5P2 = 120/(3)!

5P2 = 120/6

5P2 = 20 ways.

When it comes to combination versus permutation, in terms of combination order does matter. Order does not matter in terms of permutation. Can you explain why that is the case using a simple math example for both cases?

...
When it comes to combination versus permutation, in terms of combination order does matter. Order does not matter in terms of permutation. Can you explain why that is the case using a simple math example for both cases?

1. Why do combination and permutation have different rules for order?

Combination and permutation have different rules for order because they are two different mathematical concepts that deal with the arrangement of elements in a set. Combination is concerned with the selection of a subset of elements from a larger set, while permutation is concerned with the arrangement of all elements in a set.

2. What is the main difference between combination and permutation?

The main difference between combination and permutation is that combination does not consider the order of the selected elements, while permutation does. This means that in combination, the order of the selected elements does not matter, while in permutation, it does.

3. Can the rules for combination and permutation be used interchangeably?

No, the rules for combination and permutation cannot be used interchangeably. This is because they have different formulas and concepts, and using the wrong formula can lead to incorrect results. It is important to understand the difference between combination and permutation in order to use the correct rules for each.

4. Why is the formula for permutation n! while the formula for combination is n!/r!(n-r)!?

The formula for permutation is n!, which means n factorial, because it takes into account all possible arrangements of n elements. On the other hand, the formula for combination is n!/r!(n-r)!, which takes into account only the number of ways to select r elements from a set of n elements, without considering their order. This is why the formula for combination is a variation of the formula for permutation.

5. How do I know when to use combination or permutation in a problem?

You should use combination when the order of the selected elements does not matter, and you are only concerned with the number of ways to select a subset of elements from a larger set. You should use permutation when the order of the elements matters, and you want to find all possible arrangements of a set of elements. It is important to carefully read the problem and understand the requirements in order to determine whether to use combination or permutation.

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