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Math Help - A Simple One

  1. #1
    Junior Member SheekhKebab's Avatar
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    A Simple One

    I have a question about the following rule:-

    Number of ways of selection of r things out of n identical things is 1. (r \le n).

    Why is the number of ways = 1 ? Say there are 7 identical A's - A,A,A,A,A,A,A and we have to select 2 A's, then there are multiple ways of selecting 2 A's e.g we can select two A's from left corner or we can select 2 A's from the right corner or we can select 2 A's from middle......so on . In fact we can select 2 A's in any order and there will be multiple ways of selecting 2 A's. So why the rule says that there is only one way ? What am I missing ? Request advice on this.

    Thanks in advance.
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    Junior Member SheekhKebab's Avatar
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    Re: A Simple One

    What's the logic behind this rule ?
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    Re: A Simple One

    Quote Originally Posted by SheekhKebab View Post
    I have a question about the following rule:-

    Number of ways of selection of r things out of n identical things is 1. (r \le n).

    Why is the number of ways = 1 ? Say there are 7 identical A's - A,A,A,A,A,A,A and we have to select 2 A's, then there are multiple ways of selecting 2 A's e.g we can select two A's from left corner or we can select 2 A's from the right corner or we can select 2 A's from middle......so on . In fact we can select 2 A's in any order and there will be multiple ways of selecting 2 A's. So why the rule says that there is only one way ? What am I missing ? Request advice on this.

    Thanks in advance.
    The rule is simple common sense. I have 100 identical red balls. I choose three. What do I have as a result? Three red balls: that is the only possible answer. There is no way to differentiate between one collection of three identical balls and another collection of three identical balls. The rule might be better expressed by saying that the number of distinguishable ways of choosing r items out of n indistinguishable items
    (given - 1 < r < n + 1) is 1.

    You are confusing yourself by making your A's non-identical. You are saying that the two leftmost A's are NOT IDENTICAL to the rightmost A's by virtue of their original position. But anyone who never saw the original position would not be able to distinguish two A's selected from the left from the two A's selected from the right.

    Going back to my ball example, if each red ball was identical to the other 99 EXCEPT that the balls were numbered 1 through 100 (a situation analogous to your positions), the number of distinguishable selections would be 161,700.
    Last edited by JeffM; August 25th 2014 at 07:57 AM.
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    Re: A Simple One

    The crucial point is that all the items are identical if you had 3 different objects, a, b, and c, say, there would be 3!= 6 ways to order them:
    abc, acb, bac, bca, cab, and cba. If you have three identical items, a, a, and a, then those six all become "aaa", the only way to order them.
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    Junior Member SheekhKebab's Avatar
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    Re: A Simple One

    Hi guys,

    Thanks! That was really helpful ! I realize where I went wrong-I got confused by the word 'ways'. It appeared to me that though the selection will be one or same , but the ways to get that one selection will be many. I think the wording of the rule is a bit confusing. Should't it be better worded in this way :- The number of selections of r identical things from n identical things is 1. Rather than using the word 'ways'. I'm still a bit confused why are we using the word 'ways'.
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  6. #6
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    Re: A Simple One

    Yes. If it was stated as "distinguishable selections" it would be clearer.
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