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Math Help - combinations

  1. #1
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    combinations

    1. Two mating fruit flies each have one gene for black eyes and one for red eyes. The offspring receive one eye-colour gene from the male and one from the female.
    a) Produce a tree diagram showing all the possible combinations for the offspring.
    b) How many distinct combinations can be produced?


    2. There are 14 members in a student government. They decide to send a committee of 3 students, one of who will be designated as the spokesperson, to negotiate a new attendance policy with the principal.
    a) In how many ways can this be done?
    b) Suppose there are 8 women and 6 men in the student government. How many possibilities are there if the committee must have at least 1 person of each gender?


    3. How many 8-letter sequences are there consisting of As, Bs, and Cs with the following restrictions?
    a) The sequence must have at least one A.
    b) The sequence must have at least one of each symbol.
    c) The sequence must have exactly 4 As.
    d) The sequence must have exactly 3 As and 3 Bs.


    4. A necklace is made out of identically shaped, coloured beads. Ten beads are red, 9 are black, 7 are white, 12 are pink, and 4 are purple. The beads are strung together on a waxed string. The ends are tied together to form a large knot that prevents the beads from sliding past it. How many different necklaces can be made in this way?
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  2. #2
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    3. How many 8-letter sequences are there consisting of As, Bs, and Cs with the following restrictions?
    a) The sequence must have at least one A.
    Count the number of ways with no A's and subtract from the total number of sequences.
    There are 3^{8} sequences with A,B,C. There are 2^{8} ways with
    B and C. 3^{8}-2^{8}

    b) The sequence must have at least one of each symbol.
    Count the sequences that have NONE of each symbol and subtract . By that, I assume they mean it must contain one of each. So, all of one kind or two different ones are allowed. You could have all A's, all B's, all C's, B and C, A and B, A and C

    3^{8}-(3+3\cdot{2^{8}})


    4. A necklace is made out of identically shaped, coloured beads. Ten beads are red, 9 are black, 7 are white, 12 are pink, and 4 are purple. The beads are strung together on a waxed string. The ends are tied together to form a large knot that prevents the beads from sliding past it. How many different necklaces can be made in this way?

    There are 42 beads. So the arrangements are \frac{\frac{42!}{10!9!7!12!4!}}{2}

    We divide by two to keep from overcounting because the necklace can be turned around.


    I hope I didn't miss something.
    Last edited by galactus; July 2nd 2007 at 05:50 PM.
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  3. #3
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    May I be so impertinent as to ask if you are just asking for someone to do your homework? I must say that the inconsistence levels of the problems puzzle me: they range from trivial to very difficult.

    Here are some suggestions.
    3a 3^8 -  2^8. WHY?
    3b Surj(8,3), the number of surjections from a set of eight to a set of three.
    3c  \sum\limits_{b = 0}^4 {\frac{{8!}}{{\left( {4!} \right)\left[ {\left( {4 - b} \right)!} \right]\left( {b!} \right)}}}
    3d Completely trivial if you understand any of this!

    4 \frac{1}{2}\left[ {\frac{{42!}}{{\left( {10!} \right)\left( {9!} \right)\left( {7!} \right)\left( {12!} \right)\left( {4!} \right)}}} \right].
    The big question here is why \frac{1}{2}?
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  4. #4
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    Yes i can answer that

    To Plato


    Right now I am reviewing and studying for my grade 12 geometry exam, which believe me is no walk in the park. So to better ready myself, I'm trying to do random questions that are uncommon just so I won't be surprised in the exam. And no, I am not having people do my homework for me. I do my work and post the questions that i have a hard time with. But never the less, i really appreciate everyone who is helping and will continue to help me!

    So THANKYOU!!!
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  5. #5
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    So can you help me with the rest?

    Can anyone tell me what to do for number one and two.
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