# Understanding fundamental principles of counting.

There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication.

I often got confused applying them. I know that if there are two jobs, say m and n, such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two jobs can be performed in $m+n$ ways and when two jobs are performed in succession, they can be performed in $m\times n$ ways.

My question is how to identify whether jobs are independent or in succession?

Is there any simple way to identify this? Are there any keywords?

Thanks for helping me.

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@Arturo Dear sir Does independent refers to two different events? –  srijan May 18 '12 at 19:49
"Independent" refers to the fact that a choice associated to one set of options does not affect your available choices for the second. For example, say you go to dinner, and want to have an entree and wine with it, and you can choose from among 5 different entrees and 4 different wines. If you are a stickler for wine, your choice of entree may restrict your choice of wine (if you order fish for your entree, you cannot order red wine; if you order meat for your entree, you cannot order white wine), so the choices are not independent. (cont) –  Arturo Magidin May 18 '12 at 19:51
So in that case, the product and sum rule don't apply, because one choice affects the other. If the choice of entree does not affect your options of wine (if you don't care about choosing the "wrong wine" for your dish), then the two options are "independent", in that one decision for one of the items does not restrict or expand your available choices for the other. –  Arturo Magidin May 18 '12 at 19:52
@srijan: You have to be careful about this keyword business. We do often use such keywords on tests, so that people who don't quite know what's going on but have studied have a reasonable chance to do OK. However, real world problems do not come with keywords carefully attached. –  André Nicolas May 18 '12 at 20:23
@AndréNicolas Ok sir i will take care of that. I didn't mean that. –  srijan May 18 '12 at 20:41

The distinction is not between "independently" and "in succession". The real distinction is whether you are doing both, or you are doing just one.

The number of ways to do one of the jobs is $m+n$ (either do the first job, which can be done in $m$ ways; or do the second job, which can be done in $n$ ways; total number of ways, $m+n$).

The number of ways of doing both is $mn$, because you have $m$ ways of doing the first job, and $n$ ways of doing the second job, and you have to do both.

Say you have $5$ pants and $3$ shirts. If you are giving one piece of clothing away, then you have $5+3$ ways of deciding which piece to give away. But if you are deciding what to wear, you need to pick a pair of pants, and a shirt. That gives you $5\times 3$ possible combinations. You can make the choices simultaneously (they don't have to be "in succession"). The "independence" clause is jut that choice of shirt/pants should not restrict your choice of pants/shirt (so you don't have a plaid shirt and a pair of striped pants that cannot be worn together...). What you choose for job one does not affect what you can choose for job two.

So you have to think about what you are doing. I'm not sure if there are "keywords" that you should be looking at.

I hadn't ever realized that there was this relationship between "and/exclusive or" and numeric multiplication/addition, even though I knew that, for Boolean algebras, taking "and" and "exclusive or" as $\otimes$ and $\oplus$ makes the algebra a ring... –  Thomas Andrews May 18 '12 at 20:32