# combinatorics – \$18\$ guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side…

“My doubt is: shouldn’t we multiply by 2P1 as well because the four or the three particular guests can be seated on either side of the table?”

No, the question says:

Four particular guests desire to sit on one particular side and
three others on the other side

i.e.: four want to sit on the left side and the other three wants to sit on the right side, or: four want to sit on the right side and the other three want to sit on the left side, but not both at the same time. It’s either one scenario or the other.

In other words, if you ask the four, “where do you want to sit?” They won’t collectively answer: “I don’t mind which side, so long as we are all on the same side, and the other three are on the other side”. They will answer either, “Us four want to sit on the left side (and therefore the other three want to sit on the right side)”, or they will answer, “Us four want sit on the right side (and therefore the other three want to sit on the left side)”.