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Thread: How to know whether objects can be treated as interchangeable

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    How to know whether objects can be treated as interchangeable

    Hi,
    In questions where they ask you for the number of arrangements of (mn) number of people to be split in to groups of m people (where m and n are positive integers), how do you know whether to divide by m! (I read somewhere that it's based on whether their interchangeable, however, how do you tell??). In the question: In how many ways can 12 different presents be divided into 4 piles of 3 presents they give the answer to be (369 600)/4!, however, when the question is: In how many ways can 12 different presents be divided between 4 children, if each child is to recieve the same number of presents- they give the answer to be 369 600.

    Any help would be great
    Last edited by frog0101; 8 Oct 2017 at 7:55 AM.

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    Re: How to know whether objects can be treated as interchangeable

    Quote Originally Posted by frog0101 View Post
    Hi,
    In questions where they ask you for the number of arrangements of (mn) number of people to be split in to groups of m people (where m and n are positive integers), how do you know whether to divide by m! (I read somewhere that it's based on whether their interchangeable, however, how do you tell??). In the question: In how many ways can 12 different presents be divided into 4 piles of 3 presents they give the answer to be (369 600)/4!, however, when the question is: In how many ways can 12 different presents be divided between 4 children, if each child is to recieve the same number of presents- they give the answer to be 369 600.

    Any help would be great
    Basically if the things your dividing it into are the same, i.e. if you swap them around it won't change your number of combinations. Also each pile/child must receive an equal amount to divide by m!.

    For example the pile one you divide by 4! as there are 4 piles which are the same so after you distribute the presents you divide by 4! as the piles are the same. However with the children one, since you can also arrange the children then you wouldn't divide.

    E.g. you have 2 piles, one with an apple and one with an orange, if you swap the apple and orange you still have one pile with an apple and one with an orange so it's basically a repeat, AO and OA so dividing by 2! removes the repetitions. If Billy has the apple and John has the orange, and if they swap them, then you have a new combination with Billy with the orange and John with the apple.

    Hope this makes sense
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    Re: How to know whether objects can be treated as interchangeable

    It is not a question of whether or not the objects are distinguishable. It is a question of whether or not the groups you are dividing them into are distinguishable.

    It depends on context/common sense: In your question there is no way to distinguish between the piles UNTIL you know which people the piles are going to.
    Martin_SSEDU and pikachu975 like this.

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