Three students are seated in a row of 10 chairs for an exam

Three students are seated in a row of 10 chairs for an exam.

Assuming all outcomes are equally likely, what is the probability that they

are in three adjacent chairs? What is the probability that no two adjacent

chairs are occupied?

Solution

# ways of arrangement = ¹⁰P₃

# ways of having arrangement in adjacent = 8*3! taking those as 1 person and 8 chairs becomes the same results which also should also include internal arrangements

# ways of arranging in non. adjacent any 2 is

let those three be sited

and now no.of chairs at one end =x, middle = y,next middle =z, other end =w

so x,z>=0,y,z>=1

and x+y+z+w=8 or x+(y-1)+(z-1)+w=6.

=> # of ways = ^6+4-1C_4-1 = ⁹C₃ =84

and now including internal arrangement = 84*6 = 504

so now

(a)

the probability that they are in three adjacent chairs = 1/15

(b)

the probability that no two adjacent chairs are occupied = 504/720 = 0.7