GATE Mechanical 2021 Set 2 | GA | Question: 1

Question

Five persons $\text{P, Q, R, S}$ and $\text{T}$ are to be seated in a row, all facing the same direction, but not necessarily in the same order. $\text{P}$ and $\text{T}$ cannot be seated at either end of the row. $\text{P}$ should not be seated adjacent to $\text{S. R}$ is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Given that, five persons $P, Q, R, S$ and $T$ are to be seated in a row.

• $P$ and $T$ cannot be seated at either end of the row.
• $P$ should not be seated adjacent to $S.$ 
• $R$ is to be seated at the second position from the left end of the row.

 Now, the five persons can be seated in the below arrangements:

• First we fix the position of $R,$ and now we have two choices for $P$ and $T,$ then place the $S$ and $Q$ in the remaining positions.  
  • $\begin{array}{|c|c|c|}\hline S & {\color{Red} {R}} & {\color{Green}{P}}  & {\color{Blue}{T}}  & Q  \\\hline\end{array}$
  • $\begin{array}{|c|c|c|}\hline Q & {\color{Red} {R}} & {\color{Green}{P}}  & {\color{Blue}{T}}  & S  \\\hline\end{array}$
  • $\begin{array}{|c|c|c|}\hline S & {\color{Red} {R}} & {\color{Blue}{T}} & {\color{Green}{P}}  & Q  \\\hline\end{array}$ 

The number of distinct seating arrangements possible is $ = 3.$

So, the correct answer is $(B).$

Answer 2

Given that, five persons $P, Q, R, S$ and $T$ are to be seated in a row.

• $P$ and $T$ cannot be seated at either end of the row.
• $P$ should not be seated adjacent to $S.$ 
• $R$ is to be seated at the second position from the left end of the row.

 Now, the five persons can be seated in the below arrangements:

• First we fix the position of $R,$ and now we have two choices for $P$ and $T,$ then place the $S$ and $Q$ in the remaining positions.  
  • $\begin{array}{|c|c|c|}\hline S & {\color{Red} {R}} & {\color{Green}{P}}  & {\color{Blue}{T}}  & Q  \\\hline\end{array}$
  • $\begin{array}{|c|c|c|}\hline Q & {\color{Red} {R}} & {\color{Green}{P}}  & {\color{Blue}{T}}  & S  \\\hline\end{array}$
  • $\begin{array}{|c|c|c|}\hline S & {\color{Red} {R}} & {\color{Blue}{T}} & {\color{Green}{P}}  & Q  \\\hline\end{array}$ 

The number of distinct seating arrangements possible is $ = 3.$

So, the correct answer is $(B).$