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if n= {1,2... n} how many functions are there from n to n? Watch

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    little unsure
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    (Original post by Trackstar)
    little unsure
    Well, 1 can map to any of 1..n, 2 can map to any of 1..n, 3 can map to ...

    It's a problem in combinatorics.
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    (Original post by atsruser)
    Well, 1 can map to any of 1..n, 2 can map to any of 1..n, 3 can map to ...

    It's a problem in combinatorics.
    yes, i thought n^2 but apparently its n^n
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    (Original post by Trackstar)
    yes, i thought n^2 but apparently its n^n
    Each choice of mapping for 1,2, .., n is independent of the others. So if there are n for each, how many in total?

    Maybe consider a more general but also more specific example: how many maps are there from {1, 2} to {1, 2, 3}? We can map 1 to 1,2, or 3, and we can map 2 to 1,2, or 3 independently, so in total ...
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    (Original post by atsruser)
    Each choice of mapping for 1,2, .., n is independent of the others. So if there are n for each, how many in total?

    Maybe consider a more general but also more specific example: how many maps are there from {1, 2} to {1, 2, 3}? We can map 1 to 1,2, or 3, and we can map 2 to 1,2, or 3 independently, so in total ...
    yeah, so what i cant get my head around is that because 1 maps to all of 1,2,..n and so does 2, 3 etc and this happens n times it would just be n*n n^2
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    (Original post by Trackstar)
    and this happens n times it would just be n*n n^2
    This isn't right. If I want to count the mappings from {1,2,3} to {1,2,3,4}, then there are 4 choices for 1, 4 choices for 2, 4 choices for 3. Each of these can be made independently. So how many in total are there? Think of making a tree of combinations - how many branches would you need at each stage?
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    (Original post by atsruser)
    This isn't right. If I want to count the mappings from {1,2,3} to {1,2,3,4}, then there are 4 choices for 1, 4 choices for 2, 4 choices for 3. Each of these can be made independently. So how many in total are there? Think of making a tree of combinations - how many branches would you need at each stage?
    okay, are you saying that you then have to include all the different combinations in which {1,2,3} can map to {1,2,3,4} say 1-2, 2-3, 3-4 is one, 1-4, 2-3, 3-1 is another.
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    (Original post by Trackstar)
    okay, are you saying that you then have to include all the different combinations in which {1,2,3} can map to {1,2,3,4} say 1-2, 2-3, 3-4 is one, 1-4, 2-3, 3-1 is another.
    Actually forget the tree of combinations idea - I don't think that's a good way to represent things. However, yes, you are thinking along the right lines. Have you heard of the fundamental counting principle in combinatorics? If not, you should google it.
 
 
 
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