GATE Mechanical 2021 Set 1 | GA Question: 10

1 vote

Five persons $\text{P, Q, R, S and T}$ are sitting in a row not necessarily in the same order. $Q$ and $R$ are separated by one person, and $S$ should not be seated adjacent to $Q.$

The number of distinct seating arrangements possible is:

1. $4$
2. $8$
3. $10$
4. $16$

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1 vote

Given that, five persons $\text{P, Q, R, S, and T}$ are sitting in a row.

1. $Q$ and $R$ are separated by one person.
2. $S$ should not be seated adjacent to $Q.$

We can follow the above two conditions, we get the following sitting orders:

• First fix the position of $Q,$ and $R,$ we get,
• ${\color{Red}{Q} } \;{\color{Green}{P} }\; {\color{Blue}{R} }\; S\; T$
• ${\color{Red}{Q} }\; {\color{Green}{P} }\; {\color{Blue}{R} }\; T\; S$
• ${\color{Red}{Q} }\; {\color{Green}{T} }\; {\color{Blue}{R} }\; S\; P$
• ${\color{Red}{Q} }\; {\color{Green}{T} }\; {\color{Blue}{R} }\; P\; S$
• Again fix the position of $Q,$ and $R,$ we get,
• ${\color{Green}{P} }\;{\color{Red}{Q} }\; T \;{\color{Blue}{R} }\; S$
• ${\color{Green}{T} }\;{\color{Red}{Q} }\; P \;{\color{Blue}{R} }\; S$
• Again fix the position of $Q,$ and $R,$ we get,
• ${\color{Green}{S} }\; P\; {\color{Red}{Q} }\;T\;{\color{Blue}{R} }$
• ${\color{Green}{S} }\; T\; {\color{Red}{Q} }\;P\;{\color{Blue}{R} }$

Now, we can fix the positions interchangeably, and get the following siting orders:

• First fix the position of $R,$ and $Q,$ we get,
• ${\color{Blue}{R} }\; {\color{Green}{P} }\; {\color{Red}{Q} }\; T\; S$
• ${\color{Blue}{R} }\; {\color{Green}{T} }\; {\color{Red}{Q} }\; P\; S$
• Again fix the position of $R,$ and $Q,$ we get,
• ${\color{Green}{S}}\; {\color{Blue}{R} }\; P\; {\color{Red}{Q} }\; T$
• ${\color{Green}{S}}\; {\color{Blue}{R} }\; T\; {\color{Red}{Q} }\; P$
• Again fix the position of $R,$ and $Q,$ we get,
• ${\color{Green}{S}}\; P\; {\color{Blue}{R} }\; T\; {\color{Red}{Q} }$
• ${\color{Green}{S}}\; T\; {\color{Blue}{R} }\; P\; {\color{Red}{Q} }$
• $P\;{\color{Green}{S}}\; {\color{Blue}{R} }\; T\; {\color{Red}{Q} }$
• $T\;{\color{Green}{S}}\; {\color{Blue}{R} }\; P\; {\color{Red}{Q} }$

$\therefore$ The number of distinct seating arrangements possible $= 16.$

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

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