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Math Help - Combinations

  1. #1
    Junior Member
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    Combinations

    I can't get the last question.....

    A pizza company advertises that it has 15 toppings from which to choose (There are six vegetable toppings)

    e) how many different 3 topping pizzas have at least one vegetable?
    This is what i did:
    6nCr3 = 20 15nCr3 = 455
    455-19 = 436

    The answer in the back of the book is:371
    Help please?
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, tmas!

    A pizza company advertises that it has 15 toppings from which to choose.
    There are six vegetable toppings.

    e) How many different three-topping pizzas have at least one vegetable?

    There are 6 Vegetable toppings and 6 Others.


    There are: . _{15}C_3 \:=\:\dfrac{15!}{3!\,12!} \:=\:455 possible three-topping pizzas.


    The opposite of "at least one vegetable" is "no vegetables."

    There are: . _9C_3 \:=\:\dfrac{9!}{3!\,6!} \:=\:84 three-topiing pizzas with no vegetables.

    Therefore, there are: . 4565 - 84 \:=\:371 pizzas with at least one vegetable.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    If you want to list the pizzas with at least one vegetable,
    . . be sure you make a complete list.


    One vegetable and two Others
    , , (_6C_1)(_9C_2) \:=\:(6)(36) \:=\:216 ways.

    Two vegetables and one Other
    . . (_6C_2)(_9C_1) \:=\:(15)(9) \:=\: 135 ways.

    Three vegetables
    . . _6C_3 \:=\:20 ways.


    Therefore, there are: . 216 + 135 + 20 \:=\:371 vegetarian pizzas.

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