Sample example in combination

Hi,

There is sample example in combination which states that how many ways 9 chairs can be order in the group of 3 chairs?

Some part I have understood but in the answer it is divided by 3! My question is why divided by 3!.

REGARDS

Note by Pratik Patel
1 month, 2 weeks ago

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Could it be because they asked for the number of combinations instead of permutations? Go here for an explanation of combinations.

David Stiff - 1 month, 2 weeks ago

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Or there could be repeats. I'd have to see the problem to know for sure.

David Stiff - 1 month, 1 week ago

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There are 99 distinct chairs. How many ways are there to group these chairs into 3 groups of 3?

This is the question

Pratik Patel - 1 month, 1 week ago

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Oh, I see now. Did you mean 99 chairs? If so, then the first step, which it sounds like you understand, would be to find 9!9! (99 chairs to pick, then 88 chairs, and so on). However, we need to divide this by the number of ways we can arrange the groups of 33 themselves.

For instance, one way to arrange the chairs could be ABCABC, DEFDEF, and GHIGHI. However, if we don't divide by the number of group arrangements, we will also be counting other arrangements like GHIGHI, DEFDEF, and ABCABC, which is the same as our first one, just with the groups rearranged. We don't consider these to arrangements to be distinct, since each group is the same, just ordered differently. Thus, we have to divide by the number of ways we could arrange the groups, in our case, 3!3!.

Hope this helps.

David Stiff - 1 month, 1 week ago

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Thanks for the reply..

Suppose We have 6 people and we have to make 2 groups and each group contains 3 people. So in this case do we have to divide by 2! Or 3! And what will be the answer?

Regards

Pratik Patel - 1 month, 1 week ago

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Good question! In this case we would have to divide by 2!2!, since there are 22 groups, and there would be 2!2! ways to arrange them.

It's kind of like each time we come up with another way to distribute the people, we also have 2!2! ways to order the groups themselves. So we have to divide by 2!2! to only consider the distribution itself.

David Stiff - 1 month, 1 week ago

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What will be the answer?

Pratik Patel - 1 month, 1 week ago

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@Pratik Patel Oh, sorry. I didn't see that part of your question. The answer would be 6!2!=6543=360\frac{6!}{2!} = 6\cdot5\cdot4\cdot3 = \boxed{360} ways to arrange 66 people in 22 groups of 33.

David Stiff - 1 month, 1 week ago

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What will be the answer?

Pratik Patel - 1 month, 1 week ago

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