How To Tell When Order Matters Or Not

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I have encountered a problem involving combinatorics:
enter image description here



My solution to it was $(4cdot3cdot2)+(5cdot3cdot4)+(6cdot5cdot4)$.



The textbooks solution to it, however, was



enter image description here



I would understand the solution if order didn't matter, but I don't think, from the problem hints, that order doesn't matter.



Can someone please explain this to me? What about the problem shows that order doesn't matter? Thank you.






combinatorics discrete-mathematics soft-question permutations combinations







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  • Well, I guess if you have a pack of cards and you shuffle it, it's still the same pack of cards. Therefore the order doesn't matter.
    – Matti P.
    38 mins ago














up vote
4
down vote

favorite
1












I have encountered a problem involving combinatorics:
enter image description here



My solution to it was $(4cdot3cdot2)+(5cdot3cdot4)+(6cdot5cdot4)$.



The textbooks solution to it, however, was



enter image description here



I would understand the solution if order didn't matter, but I don't think, from the problem hints, that order doesn't matter.



Can someone please explain this to me? What about the problem shows that order doesn't matter? Thank you.






combinatorics discrete-mathematics soft-question permutations combinations







share|cite|improve this question























  • Well, I guess if you have a pack of cards and you shuffle it, it's still the same pack of cards. Therefore the order doesn't matter.
    – Matti P.
    38 mins ago












up vote
4
down vote

favorite
1









up vote
4
down vote

favorite
1






1





I have encountered a problem involving combinatorics:
enter image description here



My solution to it was $(4cdot3cdot2)+(5cdot3cdot4)+(6cdot5cdot4)$.



The textbooks solution to it, however, was



enter image description here



I would understand the solution if order didn't matter, but I don't think, from the problem hints, that order doesn't matter.



Can someone please explain this to me? What about the problem shows that order doesn't matter? Thank you.






combinatorics discrete-mathematics soft-question permutations combinations







share|cite|improve this question















I have encountered a problem involving combinatorics:
enter image description here



My solution to it was $(4cdot3cdot2)+(5cdot3cdot4)+(6cdot5cdot4)$.



The textbooks solution to it, however, was



enter image description here



I would understand the solution if order didn't matter, but I don't think, from the problem hints, that order doesn't matter.



Can someone please explain this to me? What about the problem shows that order doesn't matter? Thank you.






combinatorics discrete-mathematics soft-question permutations combinations




combinatorics discrete-mathematics soft-question permutations combinations






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edited 24 mins ago









N. F. Taussig

41.3k93253




41.3k93253










asked 41 mins ago









sup

1434




1434











  • Well, I guess if you have a pack of cards and you shuffle it, it's still the same pack of cards. Therefore the order doesn't matter.
    – Matti P.
    38 mins ago
















  • Well, I guess if you have a pack of cards and you shuffle it, it's still the same pack of cards. Therefore the order doesn't matter.
    – Matti P.
    38 mins ago















Well, I guess if you have a pack of cards and you shuffle it, it's still the same pack of cards. Therefore the order doesn't matter.
– Matti P.
38 mins ago




Well, I guess if you have a pack of cards and you shuffle it, it's still the same pack of cards. Therefore the order doesn't matter.
– Matti P.
38 mins ago










3 Answers
3






active

oldest

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up vote
2
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Usually in a selection with some constraints the order doesn't matter. But context is decisive.



This especially goes for a hand of cards, where in virtually any card game only the contents of your hand matters, not in what order you drew those cards.



That being said, it is slightly ambiguous and the problem should probably state it explicitly.






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  • Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
    – Mehness
    28 mins ago

















up vote
2
down vote













Deciding whether the order matters or not in this case is more of an English problem than a math problem. It's whether the phrases "select cards at random" and "the number of selections" refer to things where order matters or not.



In this case it turns out that it didn't matteer, but I see no way of being certain of that from the problem statement itself. Nothing in there mentions whether Grace cares which card is first, second and third, or if she only cares about which cards she ends up with.



That being said, if you were to calculate the probability of ending up with such a hand, it doesn't matter which interpretation you go with. You'll get the same answer either way. The same cannot be said if repetitions are allowed. If repetitions are allowed, and order matters, then a hand of $1111$ is as common as a hand of $1234$, while if order doesn't matter, then a hand of $1234$ is 24 times more likely than a hand of $1111$.






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    up vote
    1
    down vote













    The problem only tells you the the largest number in four cards is either 5, 6 or 7. It doesn't tell you in which position it is.



    Shuffling the same selection of four cards doesn't affect the larger value among them. So order doesn't count.






    share|cite|improve this answer




















    • How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
      – Arthur
      22 mins ago











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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote













    Usually in a selection with some constraints the order doesn't matter. But context is decisive.



    This especially goes for a hand of cards, where in virtually any card game only the contents of your hand matters, not in what order you drew those cards.



    That being said, it is slightly ambiguous and the problem should probably state it explicitly.






    share|cite|improve this answer




















    • Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
      – Mehness
      28 mins ago














    up vote
    2
    down vote













    Usually in a selection with some constraints the order doesn't matter. But context is decisive.



    This especially goes for a hand of cards, where in virtually any card game only the contents of your hand matters, not in what order you drew those cards.



    That being said, it is slightly ambiguous and the problem should probably state it explicitly.






    share|cite|improve this answer




















    • Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
      – Mehness
      28 mins ago












    up vote
    2
    down vote










    up vote
    2
    down vote









    Usually in a selection with some constraints the order doesn't matter. But context is decisive.



    This especially goes for a hand of cards, where in virtually any card game only the contents of your hand matters, not in what order you drew those cards.



    That being said, it is slightly ambiguous and the problem should probably state it explicitly.






    share|cite|improve this answer












    Usually in a selection with some constraints the order doesn't matter. But context is decisive.



    This especially goes for a hand of cards, where in virtually any card game only the contents of your hand matters, not in what order you drew those cards.



    That being said, it is slightly ambiguous and the problem should probably state it explicitly.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 36 mins ago









    orlp

    7,0181229




    7,0181229











    • Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
      – Mehness
      28 mins ago
















    • Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
      – Mehness
      28 mins ago















    Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
    – Mehness
    28 mins ago




    Comically badly worded question - particularly amusing is the phrase 'each card displays one positive integer without repetition from this set' :) it's almost like the output of a bot fed elementary combinatorics questions!
    – Mehness
    28 mins ago










    up vote
    2
    down vote













    Deciding whether the order matters or not in this case is more of an English problem than a math problem. It's whether the phrases "select cards at random" and "the number of selections" refer to things where order matters or not.



    In this case it turns out that it didn't matteer, but I see no way of being certain of that from the problem statement itself. Nothing in there mentions whether Grace cares which card is first, second and third, or if she only cares about which cards she ends up with.



    That being said, if you were to calculate the probability of ending up with such a hand, it doesn't matter which interpretation you go with. You'll get the same answer either way. The same cannot be said if repetitions are allowed. If repetitions are allowed, and order matters, then a hand of $1111$ is as common as a hand of $1234$, while if order doesn't matter, then a hand of $1234$ is 24 times more likely than a hand of $1111$.






    share|cite|improve this answer


























      up vote
      2
      down vote













      Deciding whether the order matters or not in this case is more of an English problem than a math problem. It's whether the phrases "select cards at random" and "the number of selections" refer to things where order matters or not.



      In this case it turns out that it didn't matteer, but I see no way of being certain of that from the problem statement itself. Nothing in there mentions whether Grace cares which card is first, second and third, or if she only cares about which cards she ends up with.



      That being said, if you were to calculate the probability of ending up with such a hand, it doesn't matter which interpretation you go with. You'll get the same answer either way. The same cannot be said if repetitions are allowed. If repetitions are allowed, and order matters, then a hand of $1111$ is as common as a hand of $1234$, while if order doesn't matter, then a hand of $1234$ is 24 times more likely than a hand of $1111$.






      share|cite|improve this answer
























        up vote
        2
        down vote










        up vote
        2
        down vote









        Deciding whether the order matters or not in this case is more of an English problem than a math problem. It's whether the phrases "select cards at random" and "the number of selections" refer to things where order matters or not.



        In this case it turns out that it didn't matteer, but I see no way of being certain of that from the problem statement itself. Nothing in there mentions whether Grace cares which card is first, second and third, or if she only cares about which cards she ends up with.



        That being said, if you were to calculate the probability of ending up with such a hand, it doesn't matter which interpretation you go with. You'll get the same answer either way. The same cannot be said if repetitions are allowed. If repetitions are allowed, and order matters, then a hand of $1111$ is as common as a hand of $1234$, while if order doesn't matter, then a hand of $1234$ is 24 times more likely than a hand of $1111$.






        share|cite|improve this answer














        Deciding whether the order matters or not in this case is more of an English problem than a math problem. It's whether the phrases "select cards at random" and "the number of selections" refer to things where order matters or not.



        In this case it turns out that it didn't matteer, but I see no way of being certain of that from the problem statement itself. Nothing in there mentions whether Grace cares which card is first, second and third, or if she only cares about which cards she ends up with.



        That being said, if you were to calculate the probability of ending up with such a hand, it doesn't matter which interpretation you go with. You'll get the same answer either way. The same cannot be said if repetitions are allowed. If repetitions are allowed, and order matters, then a hand of $1111$ is as common as a hand of $1234$, while if order doesn't matter, then a hand of $1234$ is 24 times more likely than a hand of $1111$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 19 mins ago

























        answered 27 mins ago









        Arthur

        104k799182




        104k799182




















            up vote
            1
            down vote













            The problem only tells you the the largest number in four cards is either 5, 6 or 7. It doesn't tell you in which position it is.



            Shuffling the same selection of four cards doesn't affect the larger value among them. So order doesn't count.






            share|cite|improve this answer




















            • How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
              – Arthur
              22 mins ago















            up vote
            1
            down vote













            The problem only tells you the the largest number in four cards is either 5, 6 or 7. It doesn't tell you in which position it is.



            Shuffling the same selection of four cards doesn't affect the larger value among them. So order doesn't count.






            share|cite|improve this answer




















            • How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
              – Arthur
              22 mins ago













            up vote
            1
            down vote










            up vote
            1
            down vote









            The problem only tells you the the largest number in four cards is either 5, 6 or 7. It doesn't tell you in which position it is.



            Shuffling the same selection of four cards doesn't affect the larger value among them. So order doesn't count.






            share|cite|improve this answer












            The problem only tells you the the largest number in four cards is either 5, 6 or 7. It doesn't tell you in which position it is.



            Shuffling the same selection of four cards doesn't affect the larger value among them. So order doesn't count.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 34 mins ago









            francescop21

            61914




            61914











            • How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
              – Arthur
              22 mins ago

















            • How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
              – Arthur
              22 mins ago
















            How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
            – Arthur
            22 mins ago





            How do you know it doesn't matter? What if she is using these cards to play a game where it matters which card is drawn first? We may be interested to know what the highest card is regardless of order, but we could still care about the order.
            – Arthur
            22 mins ago


















             

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