Seating arrangement probability
6 male students and 3 female students sit around a round table. The probability that no 2 female students sit beside each other can be expressed as , where and are coprime positive integers. What is the value of ?
The answer is 19.
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1 solution
First let us find out the number of all the possible cases where 9 people sit around a round table. Fix any of the nine seats for one person, and then distribute the remaining 8 seats to the remaining 8 people. Obviously, the number of these cases is ( 9 − 1 ) ! = 8 ! .
Next, assume for a moment that there are no female students but only the 6 male students. Then there are ( 6 − 1 ) ! = 5 ! ways for them to sit around a round table. Now, imagine that 6 empty chairs are put in between the 6 male students. Then the 3 female students will end up sitting apart from each other if you pick 3 out of the 6 chairs and let the 3 girls sit there. Thus, there are 3 ! 6 ! = 1 2 0 such ways to seat the girls. So, by the rule of product, there are 5 ! ⋅ 1 2 0 ways to arrange the students so that no two girls sit next to each other.
Therefore, the probability of seating the students such that no two girls sit next to each other is 8 ! 5 ! ⋅ 1 2 0 = 1 4 5 . Hence a + b = 1 9 .