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Math Help - Combinations and Probability

  1. #1
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    Question Combinations and Probability

    If my brother and I can invite 5 friends from a class of 15 to a party how many ways can we each pick 5 friends?

    What's the probability 6 different friends are invited?

    What's the prob my brother and I invite the same 5 people?
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  2. #2
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    Quote Originally Posted by mike View Post
    If my brother and I can invite 5 friends from a class of 15 to a party how many ways can we each pick 5 friends?

    What's the probability 6 different friends are invited?

    What's the prob my brother and I invite the same 5 people?
    Well, since this question's gone unanswered for a while, I'll stick my neck out here ....


    (a) {15 \choose 5} \times {15 \choose 5}.

    The reasoning: Each brother can choose 5 friends from 15 in {15 \choose 5} ways. So the number of ways two brothers can choose is {15 \choose 5} \times {15 \choose 5}.


    (b) \frac{ {15 \choose 5} \times {5 \choose 4} \times {10 \choose 1}}{ {15 \choose 5} \times {15 \choose 5}} = \frac{{5 \choose 4} \times {10 \choose 1} }{{15 \choose 5}} = \frac{50}{ {15 \choose 5}} = ......

    The reasoning:

    numerator = number of ways the two brothers can choose 6 different friends: one brother chooses 5 friends from the 15, then the other brother chooses 4 friends from that 5 and 1 friend from the remaining 10.

    denominator = number of ways the two brothers can choose 5 friends each = answer from part (a).


    (c) \frac{ {15 \choose 5} }{{15 \choose 5} \times {15 \choose 5}} = \frac{1}{{15 \choose 5}}.


    But since arrangements under restriction aren't my forte, you should probably wait for confirmation or refutation ......
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  3. #3
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    Just to check my answer...

    Is 15 C 5 = 3003?
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  4. #4
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    Quote Originally Posted by mike View Post
    Is 15 C 5 = 3003?
    Yes.
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  5. #5
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    Hello, mike!

    I totally agree with mr F . . .


    If my brother and I can invite 5 friends from a class of 15 to a party,
    how many ways can we each pick 5 friends?
    You have {15\choose5} ways of picking 5 friends.
    Your brother has {15\choose5} ways of picking 5 friends.
    . . Therefore, the two of you have: . {15\choose5} \times {15\choose5} \;=\;3003^2 ways.




    What's the probability 6 different friends are invited?
    You can pick any group of 5 friends . . . it doesn't matter.

    Your brother must pick 4 of your 5 choices: . {5\choose4} \:=\:5 ways.
    . . And he must pick one of the other ten friends: . {10\choose1} \:=\:10 ways.
    . . Hence, he has 5\times 10 \:=\:50 ways to add a sixth friend.

    Therefore, the probability is: . \frac{50}{{15\choose5}} \;=\;\frac{50}{3003}




    What's the prob my brother and I invite the same 5 people?

    You can pick any group of 5 friends . . . it doesn't matter.

    Your brother must pick the same 5 friends: . 1 way.

    Therefore, the probability is: . \frac{1}{{15\choose5}} \;=\;\frac{1}{3003}

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