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Oh, I see now. Did you mean 9 chairs? If so, then the first step, which it sounds like you understand, would be to find 9! (9 chairs to pick, then 8 chairs, and so on). However, we need to divide this by the number of ways we can arrange the groups of 3 themselves.
For instance, one way to arrange the chairs could be ABC, DEF, and GHI. However, if we don't divide by the number of group arrangements, we will also be counting other arrangements like GHI, DEF, and ABC, 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!.
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?
Good question! In this case we would have to divide by 2!, since there are 2 groups, and there would be 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! ways to order the groups themselves. So we have to divide by 2! to only consider the distribution itself.
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
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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.
Or there could be repeats. I'd have to see the problem to know for sure.
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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
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Oh, I see now. Did you mean 9 chairs? If so, then the first step, which it sounds like you understand, would be to find 9! (9 chairs to pick, then 8 chairs, and so on). However, we need to divide this by the number of ways we can arrange the groups of 3 themselves.
For instance, one way to arrange the chairs could be ABC, DEF, and GHI. However, if we don't divide by the number of group arrangements, we will also be counting other arrangements like GHI, DEF, and ABC, 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!.
Hope this helps.
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
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Good question! In this case we would have to divide by 2!, since there are 2 groups, and there would be 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! ways to order the groups themselves. So we have to divide by 2! to only consider the distribution itself.
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What will be the answer?
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2!6!=6⋅5⋅4⋅3=360 ways to arrange 6 people in 2 groups of 3.
Oh, sorry. I didn't see that part of your question. The answer would beWhat will be the answer?