How do I determine whether to use permutations or combinations to solve a problem? How do I distinguish whether a question should use permutations or combination, or both? To me, they look the same, but I end up using the wrong one.
When a question is given to me, I think, "Oh! It's using combinations." But it turns out to be using permutation methods instead.
Thanks in advance.
 A: This is a common question that my students ask and this is why I gave you the upvote.
Now, the key thing is order: does it matter? That's how you can tell the difference.
A permutation, in the problem-solving sense, is when the order of your objects matters. For example, consider the problem of counting how many ways a gold, silver, and bronze medal can be awarded amongst 9 competitors. The ordering, say, Michael, Ryan, Nathan is different than Ryan, Nathan, Michael. Why? In the first, Michael gets the gold and then gets the bronze in the second. These are two different situations. To answer the question I posed, it's simply $\frac{9!}{6!}$, or really $9 \cdot 8 \cdot 7$.
Now what if you don't care about medaling? What if you only want to be in the top three, and even a bronze is okay? Then the two orderings I mentioned above would be considered the same. Now order is not important, and so we have combinations. For combinations, you must also divide out the "repetitions," so to say, i.e., divide by $3!$ in this case. Notice the answer is smaller.
Other common permutation problems include making passwords and ordering books on a shelf, while combinations are tackled in poker hand problems and committee problems (in which each member has a specific position, i.e., president, secretary, etc.).
