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Math Help - choosing pizza toppings

  1. #1
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    choosing pizza toppings

    Bobby is getting ready to order his pizza at the pizza palace. He has a choice of 3 sizes of pizza. The palace also offers 9 different toppings. He has decided to order a pizza with a collection of toppings on one half of the pizza and a different collection of toppings on the other half of the pizza. How many options does he have for his pizza order? Two collections of toppings are considered to be different if they differ by at least one item.


    here is what I have so far:

    first he has 3 choices for size, second, he has 2^9 choices for the first half, and he has 2^9 choices for the other half.

    so would my answer be 3 * 2^9 * 2^9, or is this counting too many times?
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  2. #2
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    Quote Originally Posted by ihavvaquestion View Post
    Bobby is getting ready to order his pizza at the pizza palace. He has a choice of 3 sizes of pizza. The palace also offers 9 different toppings. He has decided to order a pizza with a collection of toppings on one half of the pizza and a different collection of toppings on the other half of the pizza. How many options does he have for his pizza order? Two collections of toppings are considered to be different if they differ by at least one item.


    here is what I have so far:

    first he has 3 choices for size, second, he has 2^9 choices for the first half, and he has 2^9 choices for the other half.

    so would my answer be 3 * 2^9 * 2^9, or is this counting too many times?
    Say the two sides are left and right. The problem doesn't explicitly state as such, but it is a natural assumption to consider the following two configurations equivalent:

    1) pepperoni on left and mushrooms on right
    2) mushrooms on left and pepperoni on right

    So with this assumption, your answer is too high and you would have to make some modifications.

    Edit: One other thing, it's not clear whether it's allowed to choose no toppings on a side. If this is not allowed, we would replace 2^9 with 2^9 - 1.

    Edit 2: Dang I really need to learn how to read. We have to disallow the possibility that the two sides are the same, by the problem statement. So assuming that it's allowed to choose no toppings on a side, and assuming that (1) and (2) above are actually distinct, the answer would be (3)(2^9)(2^9-1).
    Last edited by undefined; May 11th 2010 at 07:30 PM.
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    so does that mean i would take (3* 2^9 * 2^9)/2?
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by ihavvaquestion View Post
    so does that mean i would take (3* 2^9 * 2^9)/2?
    Well please read my edits in the first post since I thought of some things after hitting "Submit Reply."

    But I really am not a fan of these questions that are ambiguous, so we don't know what assumptions we should work with.

    Let me name the assumptions:

    A: Two pizzas are equivalent up to rotation.
    B: It is allowed to choose no toppings on a side.

    So this makes for four different sets of assumptions:

    C = A and B
    D = A and not B
    E = not A and B
    F = not A and not B

    I won't go through every set, let's just look and C and E.

    E is given by 3\cdot2^9\cdot(2^9-1) as I wrote in Edit 2 above. This is because once the toppings on one side are chosen, we can choose any collection of toppings for the other side except for the collection already chosen on the first side.

    C is given by \left(\frac{1}{2}\right)3\cdot2^9\cdot(2^9-1).

    Note that the two sides being different makes it easier for us, because otherwise we wouldn't be able to simply divide by 2 (can you see why?).
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