londonluddite wrote:

In how many ways can 6 people be seated at a round table if one of those seated cannot sit next to 2 of the other 5?

a) 720

b) 120

c) 108

d) 84

e) 48

This kind of language for students who are in a learning phase is a crime against learning. This is so confusing.

The question should have stated that 1 person cannot sit between 2 people. Sitting next does not mean that the person is in between.

Example of sentences: I sat next to my nephew and niece who are sitting together. I am sitting next to a couple of bushes in the garden which are adjacent to each other. My house is next to the two red painted houses on the street.

If the person does not want to sit in between two specific people.

Answer = Total number of cases - Number of cases when he is sitting between these 2 people

Total = (6-1)! [number of ways of making people sit in a circle] = 120

Number of cases when the person is sitting between the 2 specific people. Place the 3 people aside a bind them together, and now we will consider them as one person. Now, we have 3 free people + 1 hypothetical person (which is actually those 3 binded) = Total 4

Formula for circular combination (n-1)! = (4-1)! = 3!

Now the three which we binded, can also be arranged.

The person which has issues with the other 2 will be in between and the other 2 can only be arranged in 2 ways next to him.

So the total ways in which the guy with issues will be stuck between the people he does not want to is 3!*2=12.

Answer now is 120-12=108.

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