Question

There are 12 seats in a row of a cinema theatre, which is nearer to the entrance. 4 persons enter the theater and occupy the seats in that row (before their entry the seats are vacant). If \[{n_1}\] is the number of ways in which they occupy the seats, such that no two persons are together and \[{n_2}\] is the number of ways in which each person has exactly one neighbor \[{n_1}:{n_2}\] is
A.\[7:2\]
B.\[3:2\]
C.\[5:4\]
D.None of these

Answer 1

Hint: Here, we will first use the formula of permutations is \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\], where \[n\] is the total number of object and \[r\] is the number required and then assume the four persons and pair them using the given condition, \[{P_1}{P_2}:{P_3}{P_4}\]. So we will assume \[{x_1}\] be the number of empty seats before \[{P_1}{P_2}\], \[{x_2}\] be the number of empty seats between \[{P_1}{P_2}\] and \[{P_3}{P_4}\] and \[{x_3}\] be the number of empty seats after \[{P_3}{P_4}\] .Then we will use the values \[{y_1} = {x_1}\], \[{y_2} + 1 = {x_2}\] and \[{y_3} = {x_3}\] in the equation, \[{x_1} + {x_2} + {x_3} = 8\]. The use the formula of combinations, \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] to find required ratio.

Complete step-by-step answer:
Given that there are 12 numbers of seats.
If \[{n_1}\] is the number of ways in which the 4 persons occupy the seats, such that no two persons will sit together.
So we will have 8 empty seats and 9 gaps in between these persons.
We know that the formula of permutations, that is, \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\], where \[n\] is the total number of object and \[r\] is the number required.
Using this formula of permutations without repetition to find the value of \[{n_1}\], we get
\[
  {n_1} = {}^9{P_4} \\
   = \dfrac{{9!}}{{\left( {9 - 4} \right)!}} \\
   = \dfrac{{9!}}{{5!}} \\
\]
Calculating the factorials of the above fraction, we get
\[
  \dfrac{{9 \times 8 \times 7 \times 6 \times 5!}}{{5!}} = 9 \times 8 \times 7 \times 6 \\
   = 3024 \\
\]
We have given that \[{n_2}\] is the number of ways in which each person has exactly one neighbour.
Let us assume that the four persons are \[{P_1}\], \[{P_2}\], \[{P_3}\] and \[{P_4}\].
The four persons can be paired like \[{P_1}{P_2}:{P_3}{P_4}\] (say).
Let us take \[{x_1}\] be the number of empty seats before \[{P_1}{P_2}\], \[{x_2}\] be the number of empty seats between \[{P_1}{P_2}\] and \[{P_3}{P_4}\] and \[{x_3}\] be the number of empty seats after \[{P_3}{P_4}\].
Then we have \[{x_1} + {x_2} + {x_3} = 8\], where \[{x_1} \geqslant 0\], \[{x_2} \geqslant 0\], \[{x_3} \geqslant 0\].
Putting \[{y_1} = {x_1}\], \[{y_2} + 1 = {x_2}\] and \[{y_3} = {x_3}\] in the above equation, we get
\[
   \Rightarrow {y_1} + {y_2} + 1 + {y_3} = 8 \\
   \Rightarrow {y_1} + {y_2} + {y_3} = 8 - 1 \\
   \Rightarrow {y_1} + {y_2} + {y_3} = 7 \\
\]
Now we will find the number of solutions using the combinations of \[{y_1} + {y_2} + {y_3} = 7\], where \[{y_1} \geqslant 0\], \[{y_2} \geqslant 0\], \[{y_3} \geqslant 0\].
\[{}^9{C_2}\]
For each pair 2 can be selected in \[{}^4{C_2}\] ways, each pair is arranged in \[2!\] ways.
We will now find the value of \[{n_2}\] from the above values.
\[{n_2} = {}^9{C_2} \cdot {}^4{C_2} \cdot 2! \cdot 2!\]
Using the formula of combinations \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] in the above equation, we get
\[
   \Rightarrow \dfrac{{9!}}{{2! \cdot \left( {9 - 2} \right)!}} \cdot \dfrac{{4!}}{{2! \cdot \left( {4 - 2} \right)!}} \cdot 2! \cdot 2! \\
   \Rightarrow \dfrac{{9!}}{{2! \cdot 7!}} \cdot \dfrac{{4!}}{{2! \cdot 2!}} \cdot 2! \cdot 2! \\
\]
Calculating the factorials of the above fraction, we get
\[
   \Rightarrow \dfrac{{9 \times 8 \times 7!}}{{2! \cdot 7!}} \cdot \dfrac{{4 \times 3 \times 2!}}{{2! \cdot 2!}} \cdot 2! \cdot 2! \\
   \Rightarrow \dfrac{{9 \times 8}}{{2 \times 1}} \cdot 12 \cdot 2 \\
   \Rightarrow \dfrac{{1728}}{2} \\
   \Rightarrow 864 \\
\]
Substituting these values of \[{n_1}\] and \[{n_2}\] in \[{n_1}:{n_2}\] to find the ratio, we get
\[\dfrac{{3024}}{{864}} = \dfrac{7}{2}\]
Thus, the value of \[{n_1}:{n_2}\] is \[7:2\].
Hence, the option A is correct.

Note: In solving these types of questions, you should be familiar with the formula to find the permutations, \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\] , where \[n\] is the total number of object and \[r\] is the number required and combinations, \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\], where \[n\] is the total number of object and \[r\] is the number required. We will assume the ratio of the four persons carefully and use them to substitute in the equation \[{x_1} + {x_2} + {x_3} = 8\] to find the required value.