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How we have to start describing counting principles in combinatorics,what we need to apply first, addition principle or product principle ?
I have seen many starts with product principle rather than addition principle.One of the most common examples to introduce the lesson is calculating total different ways to travel from the city A to the city C via city B .Here many says if there are 3 different ways from A to B and another 2 different ways from B to C by the product principle number of total different ways is 3×2 = 6 . My issue is why can't we first describe, the involvement of addition principle in obtaining 3 in the first stage and 2 in the second stage because those values give number of mutually exclusive events.In that sense can't we argue that always you need to apply addition principle first and then product principle if necessary.
P.S. I have taken this example because couple of days ago very senior university lecturer used the same example without mentioning addition principle. Some times you may think this issue is not important in answering but I think answering and teaching are two different things and we have to be much precise in teaching which is the relevant part in this community.

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    $\begingroup$ The number $3$ is given, rather than being calculated as $1+1+1$. $\endgroup$ Aug 21, 2022 at 10:12
  • $\begingroup$ @StevenGubkin do you meant it's much better to say 1+1+1 to show there you have addition principle ? $\endgroup$ Aug 21, 2022 at 10:32
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    $\begingroup$ As someone who regularly teaches this class - I find calling this "Addition Principle" and "Multiplication Principle" overly pedantic. At some point you have to come to an intuitive understanding of how arithmetic operations correspond to counting problems, and verbalizing rules doesn't help. $\endgroup$ Aug 21, 2022 at 22:13
  • $\begingroup$ @AlexanderWoo answering and teaching can not be considered as in the same level. Without knowing the relevant terms you can just solve the problem but when you come to quality teaching I think we need to be much precise and give the correct picture to our students. $\endgroup$ Aug 22, 2022 at 1:51
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    $\begingroup$ Maybe this isn't what you're suggesting, but there's no reason why Principle X must always be applied before Principle Y. In practice, "sum of products" is a common schema in elementary combinatorics, i.e. addition is applied after multiplication. I agree that the addition principle should be explained in simple contexts before applying it to complicated situations. I usually do this by talking about events described using the conjunction "or". When the conjoined events are mutually exclusive the addition principle applies. Otherwise inclusion-exclusion is used to eliminate double counting. $\endgroup$ Aug 23, 2022 at 16:34

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It is not about whether you should use/introduce Addition Principle or Multiplication Principle first, but about making students to clearly understand that "Mutually exclusive options is Addition, Addition is mutually exclusive options" and hammering a similar mantra into their heads about Multiplication (though there it is a bit fancier). So, if somebody tries to use the Addition Principle, you should always ask them to identify the underlying mutually exclusive events, etc. Then they'll learn it eventually.

Then a decent amount of time should be spent on understanding PIE, complementary counting, and other stuff. AoPS used to have a nice series of short videos on all that and I believe they are still there.

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