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In how many ways can 5 girls (A, B, C, D, E) can sit on five chairs (1, 2, 3, 4, 5) if girl A does not want to sit on chair number 1 or chair number 2 and girl B wants to sit on 3rd or 4th chair?

This question was dealt in a lecture which says I should satisfy a "positive condition" first only then I should look at the "negative condition". The positive and negative conditions seem little blurry mathematics to me.

Here is how it was dealt with:

B can sit on 2 chairs (positive condition)

A can sit on 2 remaining chairs (negative)

and the rest of them can sit in 3! ways

So $2\cdot2\cdot3\cdot2\cdot1$ ways

But when I consider seats instead of people as reference

There are 3 ways in which chair number 1 can be filled ($\{C,D,E\}$)

There are 2 ways in which chair number 2 can be filled ($\left|\{C,D,E\}\right|-1$)

There are 3 ways in which chair number 3 can be filled ($\left|\{A,B,C,D,E\}\right|-2$)

There are 2 ways in which chair number 4 can be filled ($\left|\{A,B,C,D,E\}\right|-3$)

There are 1 ways in which chair number 5 can be filled (remaining girl)

So, $3\cdot2\cdot3\cdot2\cdot1$ (for seat number one to five respectively)

I have two questions regarding this problem. Is there some better way to think of the approach given in lectures other than positive/negative? Does this concept has any name?

Another question is why doesn't the latter method (the one considering seats instead of people) work?

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  • $\begingroup$ The second method is erroneous as it allows the possibility of a seating such as $(C, D, A, E, B)$ which does not satisfy the condition of the question $\endgroup$
    – Kelvin Soh
    Nov 23, 2015 at 15:43

3 Answers 3

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In your way of counting, if A sits in chair 3 you must have B in chair 4 so the choices are not independent. I don't see the advantage in the distinction between positive and negative conditions. The same problem could be posed that A wants to sit in $3,4, \text { or }5$ and B will not sit in $1,2, \text { or } 5$, which interchanges the positive and negative conditions. The key point is that whichever chair you select for B, there are the same number of options for A. If you seat A first, the number of options for B is not constant. You can divide it by cases, but seating B first avoids that.

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The second model doesn´t include the constraint:

  • B doesn't want to sit on 5th chair.

Regarding to the name of this concept, I think that it can be considered part of the optimization modelling.

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  • $\begingroup$ Edited. Thanks for the typo $\endgroup$
    – kuity kita
    Nov 23, 2015 at 15:51
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There is no distinction between the two types of conditions. Girl A could have said, "I want to sit on chair $3$ or chair $4$ or chair $5$." This would have been expressed as a $positive$ statement as far as English grammar goes, but not in any mathematical sense of the word.

The error comes in your attempt to count the number of ways seat $4$ can be allocated. The number of ways depends on how you allocate seat $3$. If seat $3$ has not been allocated to girl B then there is only one choice: seat $4$ must be allocated to girl B. If seat $3$ has been allocated to girl B, then there are two choices.

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