Combinatorics: 16 People Seated in a Row/Circle

[Finsfan342]

In how many ways can 16 people be seated:

A. In a row, if 4 of the 16 do not want to sit next to one another

B. In a row, if 3 of the 16 must be seated next to one another

C. In a circle, if 3 of the 16 must be seated next to one another

D. In a circle, if 4 of the 16 do not want to sit next to one another

These are my answers,

A. 16! - 13!4! - 14!3! - 15!2!, I'm not sure if I should include the 14!3! and 15!2!

B. 14!3! +15!2!, same situation as A

C. 12!3! + 13!2!, Little confused with the circle situation

D. 15! - 12!4! - 13!3! - 14!2!

 

[awkward]

In how many ways can 16 people be seated:

A. In a row, if 4 of the 16 do not want to sit next to one another

B. In a row, if 3 of the 16 must be seated next to one another

C. In a circle, if 3 of the 16 must be seated next to one another

D. In a circle, if 4 of the 16 do not want to sit next to one another

These are my answers,

A. 16! - 13!4! - 14!3! - 15!2!, I'm not sure if I should include the 14!3! and 15!2!

B. 14!3! +15!2!, same situation as A

C. 12!3! + 13!2!, Little confused with the circle situation

D. 15! - 12!4! - 13!3! - 14!2!

B. Wrap the special 3 together with duct tape and consider them to be one object. You then have 14 objects to arrange, which can be done in 14! ways. But the special 3 can be arranged in 3! ways, so there are 14! 3! ways in all.

For A, I would try combining the method of B with Inclusion/Exclusion.