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Combinations and ice cream

  1. Aug 30, 2013 #1
    1. The problem statement, all variables and given/known data
    To put it simply, there are 4 different flavors of ice cream, and 6 different toppings.

    a)how many sundaes are possible using one flavor of ice cream and 3 diff toppings?
    b)How many sundaes are possible using one flavour of ice cream and from 0 to 6 toppings?
    c)How many different combinations of flavors of 3 scoops of ice cream are possible if it is permissible to make all 3 scoops the same flavor?

    2. Relevant equations

    3. The attempt at a solution
    a)4*(4C3) = 80
    b)4*(4C0)+4*(4C1) etc up until 6.. I get a 256.
    c) This is the part I'm not so sure at...

    For each scoop, there are 256 combinations.. so therefore we look at it from the perspective of the different combinations of whether the scoops are of the same flavor or not? like

    SSS, SDD, DDD, SSD, same and different

    so 256x3! ?
  2. jcsd
  3. Aug 30, 2013 #2


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    I agree with your first two. For the third, can't you just use the counting principle? How many choices for the first scoop, then for the second, then for the third?
  4. Aug 30, 2013 #3
    Oh, so its 4x4x4? since the same flavors are allowed.
  5. Aug 30, 2013 #4


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    Yes, that's all there is to it.
  6. Aug 31, 2013 #5
    For (a) and (b) look again at the number of toppings. You should have spotted this when you tried to calculate 4C6.
  7. Aug 31, 2013 #6
    hmm not sure why i wrote 4 instead of 6 lol.. but yeah i wrote 6 on my work.. sry for the confusion
  8. Aug 31, 2013 #7
    But using the counting principle for c) will distinguish between the same combinations; AAB will be different than ABA etc. The counting principle is used when the order matters, i.e. permutations.

    Say your flavours A, B, C and D. You can represent the combination AAA by ***|||, the combination AAB by **|*||, ABC by *|*|*|, BDD by |*||** etc... Can you use this to count the number of combinations of 3 flavours with repetition allowed?
  9. Aug 31, 2013 #8


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    Yes, I think you are correct. I must have been thinking ice cream cones instead of sundaes in my response. :redface: Altogether too much sugar I guess.
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