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Math Help - [Probability] Easy problem but I can't figure it out! (drawing cards from a deck)

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    [Probability] Easy problem but I can't figure it out! (drawing cards from a deck)

    Suppose I have a deck with 40 cards.
    3 cards type A,
    2 cards type B,
    3 cards type C,
    and the rest from other types.
    I'll take 6 cards from this deck. What's the probability that, at the same time, I'll get at least one card of each types A, B and C?
    I know that the hypergeometric distribution should be used. I simulated 5000 times on Excel and reached an aproximated result of 0.032 but I couldn't get to this result analytically.
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    Re: [Probability] Easy problem but I can't figure it out! (drawing cards from a deck)

    Quote Originally Posted by vasantonio View Post
    Suppose I have a deck with 40 cards.
    3 cards type A,
    2 cards type B,
    3 cards type C,
    and the rest from other types. I couldn't get to this result analytically.
    Notation: Let X be the event of getting at least one of type X and X^c is none of type X.

    \mathcal{P}(A\cap B\cap C)=1-\mathcal{P}(A^c\cup B^c\cup C^c).
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