KVPY2017MathematicsPermutation and Combination
Thirty two persons X₁, X₂, , X₃₂ are randomly seated around a circular table at equal intervals. Two persons X_ i and X_ j are said to be within earshot of each other if there are at most three persons between them on the minor arc joining X_ i and X_ j . The probability that X₁ and X ₂ are within earshot of each other is, ( . Here . ( array l n r array )= n ! ( n - r ) ! r ! )
Options
- A( array c 32 2 array ) 30 ! 8(32 !)
- B( array c 32 2 array ) 30 ! 4(32 !)
- C8 31
- D4 31
Correct answer
C. 8 31
Step-by-step solution
Case - 1 No person between X ₁ & X ₂ 30 ! 2 ! 31 ! = 2 31 Case-2 One person between X₁ & X₂ ³⁰ C ₁(29) ! 2 ! 31 ! = 2 31 Case-3 When 2 person between X ₁ & X ₂ ³⁰ C ₂ 28 ! 2 ! 2 ! 31 ! = 2 31 Case-4 When 3 person between X₁ & X₂ ³⁰ C ₃ 27 ! 2 ! 3 ! 31 ! = 2 31 Total = 8 31