Question

There are five students S₁, S₂, S₃, S₄ and S₅ in a music class and for them there are five seats R₁, R₂, R₃, R₄ and R₅ arranged in a row, where initially the seat R_i is allotted to the student S_i, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

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Linked Question 1

The probability that, on the examination day, the student S₁ get the previously allotted seat R₁, and NONE of the remaining students gets the seat previously allotted to him/her is

$\frac{1}{5}$

$\frac{3}{40}$

$\frac{5}{40}$

$\frac{7}{40}$

(i) $D_{4} = 4! (\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}) = 12 - 4 + 1 = 9$

$P (E) = \frac{9}{5!} = \frac{9}{120} = \frac{3}{40} .$

S₄ → 3 ways as in S₂

S₅ → 2 ways as in S₁

$∴$ Total 14 ways.

Hence required probability = $\frac{14}{5!} = \frac{7}{60}$

Linked Question 2

For i = 1, 2, 3, 4, let T_i denote the event that the student S_i and S_i+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T₁ $\cap$ T₂ $\cap$ T₃ $\cap$ T₄ is

$\frac{7}{60}$