GATE Civil 2021 Set 2 | GA Question: 5

Question

Four persons $P, Q, R$ and $S$ are to be seated in a row. $R$ should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:

• A. $6$
• B. $9$
• C. $18$
• D. $24$

Answer 1

It is given that, four persons $P, Q, R$, and $S$ are to be seated in a row. $R$ should not be seated at the second position from the left end of the row.

Now, we can make the possible arrangements.

• From the left end, the second position can be filled in $3$ ways.
• From the left end, the first position can be filled in $3$ ways.
• From the right end, the second position can be filled in $2$ ways.
• From the right end, the first position can be filled in $1$ way.

$$\boxed{\underset{1}{3}\quad \underset{2} 3 \quad\underset{3} 2 \quad \underset{4}1}$$

$\therefore$ The number of distinct seating arrangements possible $ = 3 \times  3 \times 2 \times 1 = 18.$

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

Comments

Method 1 :-

Total places to fill = 4

2nd position from left not contain R. So R left with 3 places where we can fill.  =   C(3,1)

now rest can be filled in 3! ways.

final answer = C(3,1) * 3!

                    = 3 * 6 =18

 

Method 2 :-

Total 4 persons and we have to fill in 4 places = 4! Ways

Now R is in 2nd position then rest can be filled in = 3! ways

final answer = total ways – R is taking second place from left

                    = 24 – 6 = 18

 

 

Answer 2

Here we have 4 people and also 4 places .

Total possible ways are 4!

Now R shouldn’t be at the 2^nd Position from Left Side.

Now, _, R ,_,_ 

Three places and three people still left to be assign 

Those three people can be placed in this Three places in 3P3 = 3! ways 

There fore required answer = 4!-3!