I have encountered a problem involving combinatorics:
My solution to it was $(4\cdot3\cdot2)+(5\cdot3\cdot4)+(6\cdot5\cdot4)$.
The textbooks solution to it, however, was
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.
Best 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 order didn't matter, 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 $1111$.