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Math Help - 20 rolls of a die chance all numbers come up?

  1. #1
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    20 rolls of a die chance all numbers come up?

    If we roll a die 20 times what are the chances that all numbers 1 through 6 come up?

    I thought the size of the space was 6^20 and then the size of the event would be
    (6!*6^12) however this is obviously wrong.

    Thanks for any help
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  2. #2
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    Re: 20 rolls of a die chance all numbers come up?

    Can you compute the chances that at least one number does not come up, using derangements?
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  3. #3
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    Re: 20 rolls of a die chance all numbers come up?

    Quote Originally Posted by hmmmm View Post
    If we roll a die 20 times what are the chances that all numbers 1 through 6 come up?
    I think that reply #2 means use inclusion/exclusion.
    Lets play the "back-of-the-book" game.

    You look up the answer and see \sum\limits_{k = 0}^5 {\left( { - 1} \right)^k \binom{6}{k}\left( {\frac{{6 - k}}{6}} \right)^{20} } .
    WHAT THE HECK????
    Now it is your responsibly the explain why that answer works.
    Can you tell us all what is going on there?
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  4. #4
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    Re: 20 rolls of a die chance all numbers come up?

    Ah ok so If it as all the ways you can choose the 20 with all 6 die thrown then that is the number of surjections from the set of 6 elements to the set of 20 elements which is

    \Sigma^{5}_0 (-1)^k 6Ck (6-k)^{20}.

    We then simply divide by the total space 6^20 and arrive at what you have there.

    Sorry quite simple I was just thinking of it in a completely and much less helpful way.

    thanks a lot (sorry for so much editing I made loads of typo's)
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  5. #5
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    Re: 20 rolls of a die chance all numbers come up?

    Very good: count the surjections. That is good.
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