Number of ways to pick pens from a box How many ways are there to pick 3 pens from a box that contains 2 red pens, 1
green pen, 1 blue pen, and 1 purple pen? Here, assume that order matters.?
I've gathered that these objects are distinguishable but a solution of C(5, 3) doesn't give me what I expect. I'm assuming that the trick here is figuring out how to handle the fact that there are 2 red pens. I've browsed and understand how to do it when the problem states "how many ways are there to pick a specific colors" but I can't translate it to this one.
 A: This is a good small problem to think about techniques, because it's simple enough to solve directly.
So the direct solution could be seen as a network:

where the branches on the first two decision layers represent the ways to select a red pen or a non-red pen, and the squares at the bottom are the multiplied-up choices along that branch, giving a total of $12+9+12=33$ options of order choices from the box. Of course, once you pick a red pen, there are no special cases.
Alternatively you could start by looking at the ordered choices three from of five different-colour pens, $5 \times 4 \times 3 = 60$ and then identify how many of those have the given pens which you are now regarding as the same colour. Fortunately whether you select one or two of the red pens, the action is the same - the cases are overcounted by a factor of two (substitution or reordering). This only leaves the case where neither of the red pens are picked as distinct, which accounts for $3!=6$ options. So the result after "identicalizing" the red pens is $(60-6)/2 +6 = 27+6 = 33$ options, again.
