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Math Help - Putting letters into envelopes

  1. #1
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    Putting letters into envelopes

    A secretary randomly put 10 letters in 10 envelope
    What is the probability that at least one letter was put in the correct envelope?

    I solved it as follows

    The probability of 0 letters in the correct envelope is
    9/10 8/9.... = 9!/10!

    what is my mistake ?
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  2. #2
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    Quote Originally Posted by Hitman6267 View Post
    A secretary randomly put 10 letters in 10 envelope
    What is the probability that at least one letter was put in the correct envelope?
    You want a derangement of ten items.
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  3. #3
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    I didn't understand how to apply the concept described in the wikipedia page.
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  4. #4
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    Suppose we arrange the envelops in strict alphabetical order.
    Now randomly order the letters. There are 10! ways to do that.
    From that reference there are d_{10} arrangements where no letter is in its correct alphabetical position. You are asked to find the probability that at least one is in the correct position.

    {\displaystyle 1-\frac{d_{10}}{10!}.
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    Is there any other way to resolve this ? Because we have never used such a formula.
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  6. #6
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    Quote Originally Posted by Hitman6267 View Post
    Is there any other way to resolve this ? Because we have never used such a formula.
    No other way that I am aware of.
    That is a simple application of the inclusion/exclusion rule on ten objects.
    I don’t understand why you were asked to do this problem without having been given the tools necessary to solve it.
    Have you done the inclusion/exclusion rule?
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  7. #7
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    Yes I have been taught the inclusion/exclusion rule
    P(A U B U C)= P(A) + P(B) + P(C) - P(AC) - P(AB) - P(BC) + P(ABC)

    where does that come in ?
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  8. #8
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    Quote Originally Posted by Hitman6267 View Post
    Yes I have been taught the inclusion/exclusion rule
    P(A U B U C)= P(A) + P(B) + P(C) - P(AC) - P(AB) - P(BC) + P(ABC)
    Well use it for ten letters not just three.
    A means that letter A is in the correct envelope.
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