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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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Migrated from GO Mechanical 2 years ago by gatecse

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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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