What's the logic behind this rule ?
I have a question about the following rule:-
Number of ways of selection of r things out of n identical things is 1. (r 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.
(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.
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.
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'.