Question

PARAGRAPH "A"
There are five students $S_1,S_2,S_3,S_4$ and $S_5$ in a music class and for them there are five seats $R_1,R_2,R_3,R_4$ and $R_5$ 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.
Question:For $i=1,2,3,4,$ let $T_i$ denote the event that the students $S_i$ and $S_{i+1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_1\cap T_2\cap T_3\cap T_4$ is

• A. $\frac{1}{15}$
• B. $\frac{1}{10}$
• C. $\frac{7}{60}$
• D. $\frac{1}{5}$

Answer 1

Answer: (C)

$n\left(T_1\cap T_2\cap T_3\cap T_4\right)=Total-n\left(\overline{T_1}\cup \overline{T_2}\cup \overline{T_3}\cup \overline{T_4}\right)$
$=5!-\left({}^4{C}_{1}4!2!-\left({}^3{C}_1.3!2!{+}^3{C}_{1}3!2!2!\right)+\left({}^2{C}_{1}2!2!{+}^4{C}_1.2.2!\right)-2\right)$
$=14$
Probability $=\frac{14}{5!}=\frac{7}{60}$