I am learning permutations and combinations in school, and there is something confusing me about the addition and multiplication principle. There is a specific situation where I'm not sure if I should use multiplication or addition principle.
My understanding of why we need multiplication in combinations and permutations is like this. If we have 3 cards, and we need to choose 3 of them, on the first pick, we have 3 possible cards we can draw. On the second pick, in each of those 3 possible cards, there can be another 2 possible cards. So there are 3 sets of 2, $3 * 2$. That's why we use multiplication.
But I'm confused why we shouldn't use it in a situation like "What is the total number of cards we can pick if in 1 scenario, we can pick 3 cards out of 3 cards, and in another scenario, we can pick 1 card, out of 3 cards".
We know for picking 3 cards out of 3 cards, there can be 6 possibilities. I know then that the total possibilities should then be $6+3=9$. But I don't know why. Because following the logic of above, in each of the 6 possibilities, there can be another 3 possible cards. In a draw of 3 cards out of 3, there are 6 possibilities. Then, when we can draw 1 card out of 3, in each of those 6 possibilities, there can be another 3. So shouldn't the number of possibilities be $6*3=18$?
I know this can't be right because on that logic, if the possibility for the first event occurring is 0, then the total number of possibilities would also be 0 because of multiplication. For instance, take the question ""What is the total number of cards we can pick if in 1 scenario, we can pick 4 cards out of 3 cards, and in another scenario, we can pick 1 card, out of 3 cards". The number of possibilities for the first 1 is 0, and the second one is 3. On my logic, the total number of possibilities is 0. But that can't be right, because the total number of possibilities is 3.
But I still don't really get why the logic for multiplication in this situation is wrong, and why the logic for addition in this situation is right. All I understand is that multiplication will produce results that don't make sense, and addition will produce results that make sense. But can someone explain to me in this specific situation, why that is?
Also, can you not give "because for separate possibilities, you add not multiply" as an answer? Because just stating this principle is not going to help me. I'm trying to understand why it's true, and stating that such a principle exists isn't very helpful.
Thanks in advance.
Best Answer
From what I can understand from your description, it depends on the question being asked. For multiplication to be appropriate for the scenarios you describe the question would be, "how many ways can you select (order) $3$ cards followed by a random selection of one of the $3$ cards?" In this case there would be $6\times 3 = 18$ ways to do that as there would be $18$ four cards sets having selected one of the cards twice.
For addition, the question would be "how many ways can you select (order) $3$ different cards plus how many ways can you select one card from the $3$?" In this case, we have $6$ three card sets plus $3$ one card sets making a total of $9$.