Seven people are seated randomly.what is the proabability that a designated two people .......................

Question

Seven people are seated randomly.what is the proabability that a designated two people sit next to each other if the seven people a)sit in a row
b)sit at a circular table?

Answer 1

To solve this, think of the two people as one. Then the probability they are next to each other in a line is 2*6!/7! = 2/7. The probability they are next to each other around a table is 2*5!/6! = 2/6 = 1/3, (the number of ways of ordering the two people times the number of seating 6 people around a table divided by the number of ways of seating 7 people around a table).

Answer 2

The number of unrestricted permutations is 7!.

The number of permutations where two designated people are next to each other:

a) in line is (2!*6)(5!)
Probability = (2!*6)(5!) / 7! = 2*6!/7! = 2/7

a) in a circle is (2!*7)(5!)
Probability = (2!*7)(5!) / 7! = 2*5!/6! = 2/6 = 1/3