Question

A team represents their school in a quiz competition. The team consists of 6 students and 3 mentors. A seating arrangement is made by placing nine chairs for this team in a row. The mentors arrive before the students and choose their seats so that each mentor ends up seating between two students. In how many ways can the mentors choose their seats?

Note: If your answer for any blank is 66,666 then put 66666 in the blank. Avoid commas and unnecessary spaces for your resulting number.

Answer 1

here we have nine seats and each mentor sits between two students

so the mentors cannot sit at ends of the chairs ie they cannot sit at extreme ends

hence they have got only 7 seats to chose from

lets say there is only one student between mentors

ie arrangement is like

S M S M S M S S S in this arrangwmwnt basically we get 4 seats for Mentors

and we can chose any three in 4C3 ways ie 4 ways

and the mentors can also interchange their positions so total permutations are 6

hence total ways in this method are 24

now lets say we have students in between our mentors

S M S S M S S M S

in this way we have got 3 seats for mentors and 3 mentors to sit hence combinations are 3C3 = 1

and permutations are 3*2 = 6 [because mentors can interchange their seats]

hence total ways in this method are = 6

hence total ways number of ways are 24+6 = 30

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