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. $16$

Comments

ans=16(option d)

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 $(D).$

Answer 2

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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Can someone confirm if below method is correct @Lakshman Bhaiya :.

Total ways possible when Q and R have one person in between them = 36

(consider 'Q_R' as one unit,therefore ways to arrange 3 units i.e. '_','Q_R','_' = 3! but now Q and R can be interchanged and 'P' OR 'S' OR 'T' can come in between Q and R ,therefore total cases possible = 3! * 2 * 3 = 36)

Out of total 36 ways, eliminate those arrangements where S and Q are adjacent,thereofore ways where S and Q are adjacent = 4 + 4+ 12 = 20

Final ans = 36 - 20 = 16