GATE Mechanical 2021 Set 1 | GA Question: 10

Question

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:

• A. $4$
• B. $8$
• C. $10$
• D. $16$

Answer 1

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 $(D).$

Comments

instead of listing them, we can use permutations formulas.

i mean, in case 1 : Q is at extreme left,

we can fill the middle place by $\binom{2}{1}$ due to either P or T can be filled.

after R, it is like 2x1 permutation.

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@Lakshman Patel RJIT sir, the rendering is not right.

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time taking approach.