a) a king and a queen
b) a king or queen
c) king and a red card
d) king or a red card
For a), I can see that, since there are 4 kings and 4 queens in a deck, the number of ways of picking both would be $(4)(4) = 16$. This is the multiplication rule? Why does this work, and how would I know to use it?
For b), I can see that the ways of picking a king OR a queen is, instead of multiplying, I must add $4+4=8$ ways. Addition rule? When does this work, and how do I know to use it?
c) and d) are where things start to get even fuzzier for me. How would I go about starting these?
I'm thinking that if a) is similar to c), then couldn't I use the multiplication rule? As in, 4 kings times 26 total red cards? But this produces an incorrect answer. Next, I thought that maybe I should exclude the kings from the set of red cards, but 4 times 22 is still incorrect.
Best Answer
The reason you can use the multiplication rule in the first one is that you are computing the number of ways to take one item each from two sets which are disjoint. There may be a more elegant way of explaining this, but I am still an amateur. :)
Presumably, by "a king and a red card" they mean "two cards, one of which is a king, and the other of which is a red card." E.g. a single red king would not suffice. If this is the case, you must take $2(26)$, which is 2 for the black kings and 26 for the red cards (same rule as before). Now you start counting the cases involving sets that are not disjoint. Take the king of diamonds...there are 25 red cards you can pair with it. Now take the king of hearts. Again there are 25 red cards you can pair with it, but one of the possibilities (drawing the two red kings) was already considered. So the answer should be $$2(26) + 1(25) + 1(24).$$ In part d, you just have to take the number of elements in the set $\{\text{King or red card}\}$, which is 28.
Edit: I fixed an error, but then my correction was wrong, so I changed it back!