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2 solutions
Let the no. of ways of arranging n distinct objects = Tn .
It's well known that Tn+1=(n+1)Tn.T1=1 is quite trivial.
Plugging n=0,one can get T0=T1=1.
That's merely an algebraic approach.I'll have to think a lot to explain it logically.
But that's a nice question,liked it.
I don't think that you can go directly by the formula.
Like - The number of ways of choosing 150 objects from 3 is 0. But according to the formula , it is 3 c 150. which makes it undefined.
SO , you can't always abide by the formula...SO , I think that it should be undefined, because there is no object , what you will arrange?? Lets hear it from some other people also to clarify.
@Calvin Lin @Pi Han Goh @Chew-Seong Cheong @Md Zuhair @Rahil Sehgal
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Dont you think Philosophically, the number of ways to arrange nothing is only by keeping nothing thus 1 . way.
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I agree to it a bit, but still , what do you mean by arranging nothing?? Its something strange. I am not sure about it.
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@Ankit Kumar Jain – Okay, Let, @Calvin Lin @Chew-Seong Cheong , @Pi Han Goh , @Brian Charlesworth , @Rahil Sehgal , @rajdeep brahma , @Brilliant Mathematics sirs ,give their thoughts about this one
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@Md Zuhair – Mathematically,
4 ! = 5 5 ! , 3 ! = 4 4 ! , 2 ! = 3 3 ! , 1 ! = 2 2 ! , 0 ! = 1 1 ! = 1 .
Since this pattern yields a definite value for 0 ! , if we acknowledge the formula and interpretation for n ≥ 1 , then why shouldn't we acknowledge it for n = 0 ? If we try to extend it to ( − 1 ) ! we would have the undefined 0 0 ! , so the pattern stops at 0 .
Philosophically, there is one way to not arrange something, namely to not arrange it. So there is one way to arrange nothing.
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@Brian Charlesworth – Thankyou sir!
@Brian Charlesworth – @Brian Charlesworth sir, Why dont we take " one way to not arrange something, namely to not arrange it" in our regular problems and add +1 to the solutions?
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@Md Zuhair – In our "regular" problems we seek to arrange things, i.e., to not n o t arrange them, so the one way to not arrange them is not part of the solution to regular problems.
What happens with 0 objects is an interesting thought process. I have inferred a numerical equivalence between the number of ways of "not arranging something" and "arranging nothing". I'm not sure that they are logically equivalent, though. I'll have to think some more about that.
@Md Zuhair – Lets wait.:) :)
Simple. There is only one way of arranging 0 objects. That is to leave the objects as they are. In fact, this is one of the reasons of why 0 ! = 1
No. of ways to arrange n objects is n ! and hence for no objects it is 0 ! = 1 .