Principle of Inclusion Exclusion regarding circular arrangements

In summary, the problem is asking for the number of ways to seat two biologists, two chemists, and two physicists at a round table with 6 chairs so that no two scientists of the same type are seated next to each other. The solution uses the Principle of Inclusion-Exclusion to calculate the total number of seatings with no restrictions, subtracts the number of seatings with 1 pair adjacent, and then adds back the number of seatings with 2 pairs adjacent. This is because the initial subtraction counts some seatings multiple times. The second step of adding back the number of seatings with 2 pairs adjacent accounts for these repeats. The convention in mathematical discussions is that "There are N things" means "There
  • #1
drifter24
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I've been working through the AOPS Intermediate Counting and Probability text, but do not understand the explanation given for one of the problems.

Two biologists, two chemists and two physicists sit at a round table with 6 chairs. In how many ways can they sit so that no two scientists of the same type are seated next to each other?

I don't understand the given PIE (Principle of Inclusion Exclusion) solution.
According to the book,
# of seatings with no pair adjacent = # of seatings (with no restriction) - # of seatings with 1 pair adjacent + " 2 pair adjacent - " 3 pairs adjacent

I don't understand why 2 pair adjacent is being added to the problem. If we eliminate with all seatings with 1 pair adjacent, does that not mean we also eliminate all invalid options? Isn't it true that there cannot be a valid option/arrangement in the set of (1 pair adjacent)--no matter how you arrange the other 4 scientists once you have 1 pair adjacent in the set--everything is invalid. I see how the solution makes sense in other examples, but the idea of a circular table is throwing me off tremendously. I can't seem to create concrete examples of repeats, when I try to draw out the tables and subsequently position the people.

Could someone please explain the second step (1 pair and 2 pair)? Thanks.
 
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  • #2
The number of seatings "with 1 pair adjacent" probably means the the number of seatings "with at least 1 pair adjacent", so it counts the seatings with 2 or more pairs adjacent.

The convention in most mathematical discussions is that the statement "There are N things..." means "There are N or more things...". If you want there to be exactly N things, you are supposed to say "There are exactly N things...".
 

FAQ: Principle of Inclusion Exclusion regarding circular arrangements

What is the Principle of Inclusion Exclusion regarding circular arrangements?

The Principle of Inclusion Exclusion is a combinatorial principle used to count the number of elements in a finite set that satisfies certain conditions. Specifically, it is often used in circular arrangements to determine the number of ways to arrange objects in a circle while considering certain constraints.

How does the Principle of Inclusion Exclusion work?

The Principle of Inclusion Exclusion states that the number of elements in a union of two or more sets is equal to the sum of the individual sets minus the number of elements in the intersection of those sets. In the context of circular arrangements, this means that we can calculate the number of ways to arrange objects in a circle by considering the number of ways each object can be placed individually, subtracting the number of ways they can be placed together, and adding back the number of ways they can be placed in groups of three, four, and so on.

When is the Principle of Inclusion Exclusion most commonly used in circular arrangements?

The Principle of Inclusion Exclusion is often used in circular arrangements when there are constraints on how the objects can be arranged. For example, if we want to arrange n objects in a circle with k objects always adjacent to each other, the Principle of Inclusion Exclusion can be used to determine the number of possible arrangements.

What is the formula for the Principle of Inclusion Exclusion in circular arrangements?

The formula for the Principle of Inclusion Exclusion in circular arrangements is given by: N = n! - (k * (n-1)!) + ((k^2) * (n-2)!) - ((k^3) * (n-3)!) + ... + (-1)^r * ((k^r) * (n-r)!) + ... + (-1)^(n-1) * ((k^(n-1)) * 1!), where n is the number of objects to be arranged, k is the number of objects that must be adjacent to each other, and r is the number of objects that must be adjacent to each other in a group.

Can the Principle of Inclusion Exclusion be applied to non-circular arrangements?

Yes, the Principle of Inclusion Exclusion can be applied to non-circular arrangements as well. It is a general combinatorial principle that can be used to count elements in a finite set satisfying certain conditions, regardless of the shape or arrangement of the set. However, it is most commonly used in circular arrangements due to its ability to handle constraints on adjacent objects.

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