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Using the inclusion-exclusion principle the number of arrangements can be 1080069120.
What is The principle of inclusion and exclusion?
- (PIE) is a counting method that determines how many elements fulfill at least one of numerous properties while making sure that components that satisfy several properties are not tallied more than once.
- Let us denote by R, the set of arrangements where the Russians are together, by A, the set of arrangements where the Americans are together, and by C for the Chinese.
I denote the complement of a set X by X' its cardinality by ∣X∣ and the universal set by S
The number of unrestricted arrangements of these 11 people.
- =11!=S
Americans can seat in:
A = 9! * 4!
Russians can seat in:
- R = 10!*3!
And Chinese can seat is:
- C = 8! *5!
Now, we calculate ∣A∩R∣. In this case,
- ∴∣A∩R∣=7!*4!*3! and for |R∩C|=3!*5!*8! and |A∩C|=4!*5!*9!
- ∴∣C∩R∩A∣=3!×4!*5!*3!
By the principle of inclusion and exclusion, we have:
- ∣C∪R∪A∣=∣C∣+∣R∣+∣A∣−∣C∩R∣−∣C∩A∣−∣R∩A∣+∣C∩R∩A∣
- or,
- ∣C'∩R'∩A'∣= S-∣C∪R∪A∣ = 1080069120
Therefore, using the inclusion-exclusion principle the number of arrangements can be 1080069120.
Learn more about the inclusion-exclusion principle, here:
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