Question: Four boys and four girls are randomly seated in 8 adjacent seats at a theatre. If it is found that a...

Question

Four boys and four girls are randomly seated in 8 adjacent seats at a theatre. If it is found that all the girls are seated in 4 adjacent seats, then the probability that the four boys are also in 4 adjacent seats is:
A. $\dfrac{1}{4}$
B. $\dfrac{3}{4}$
C. $\dfrac{1}{2}$
D. $\dfrac{2}{5}$

Answer 1

Solution

Hint : In order to solve this question we will first find the number of ways that the girls can sit together with then we will choose the number of ways in which we will make the boys to sit together with along with the girls sitting together and we will put that total outcomes to the favorable outcomes.

Complete step by step solution:
For solving this question we will first get the total outcomes for this let me explain in you in simple manner for this let there are 8 seats like this:
${\text{- , - , - , - , - , - , -, -}}$
So these are the 8 empty seats which we have to fill in such way that all the girls sits together with so:
1st way to sill all the girls together:
${\text{G G G G - , - , - , -}}$
The 2nd way to sit all the girls together:
${\text{- , G G G G - , -, -}}$
The 3rd way to sit all the girls together:
${\text{-, -, G G G G -, -}}$
The 4th way to sit all the girls together:
${\text{ -, -, - , G G G G -}}$
The 5th way to sit all the girls together is:
${\text{- , - , -, - , G G G G}}$
So the total number of ways to sit all the girls together is 5:
So out total outcomes will be 5
Now as we can see that in 1st and 4th way we can make sit all the boys together along with the all the girls sitting together:
${\text{B B B B G G G G}}$
This is one way.
${\text{G G G G B B B B}}$
And this is the other way.
So the favorable outcomes will be 2.
So the probability will be $\dfrac{2}{5}$
So the correct option will be D.
So, the correct answer is “Option D”.

Note : While solving these types of problems we should keep in mind that we have to go step by step without making any hurry. It is because we can miss one of the total outcomes from five if we are not attentive. And like this we will get the probability equal to $\dfrac{1}{2}$ .