Graham Beasley

2022-07-19

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?

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?

bgr0v

Beginner2022-07-20Added 13 answers

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 5×3 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.

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 5×3 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.

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