Question

7 boys and 3 girls are seated in a row randomly. The probability that no boy sit between two girls is
A.$\dfrac{{7! \times 3!}}{{10!}}$
B.$\dfrac{1}{{15}}$
C.$\dfrac{{8!}}{{10!3!}}$
D.$\dfrac{{8!}}{{10!7!}}$

Answer 1

Here we have to find the probability that no boy sits between two girls. We will first arrange the 7 boys and 3 girls. We will represent the positions of boys by $B$ and the position of girls by $G$. We will find the total number of ways in which we can arrange all 7 boys and 3 girls and then we will find the number of ways in which no boy sits between two girls. The required probability will be the ratio of the number of ways in which no boy sits between two girls to the total number of arrangements.

Complete step by step solution:
We will first find the total number of ways in which seven boys and three girls can be arranged.
Therefore,
Number of ways in 7 boys and 3 girls can be seated without any restrictions$= {}^{10}{P_{10}}$
We know the formula;
${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
Therefore,
${}^{10}{P_{10}} = \dfrac{{10!}}{{\left( {10 - 10} \right)!}} = \dfrac{{10!}}{{0!}}$
We know, the value of zero factorial is 1.
Therefore,
${}^{10}{P_{10}} = \dfrac{{10!}}{{0!}} = 10!$
Thus, number of ways in 7 boys and 3 girls can be seated without any restrictions$= 10!$
We can arrange the 7 boys and 3 girls as:-
${\text{B B B B B B B G G G}}$
Here B is the position of a boy and G is the position of a girl.
We will consider the position of three girls as one position.
Thus, the number of ways in which 7 boys can be arranged $= 8!$
All girls are arranged consecutively but the position of girls can be interchanged.
Number of ways in which 3 girls can be arranged at three places $= 3!$
Thus, the possible arrangement in which no boy sits between any two girls$= 8!.3!$
Hence, the required probability that no boy sit between two girls is$\dfrac{{8!3!}}{{10!}}$
On evaluating the factorial of numbers in numerator and denominator $= \dfrac{{8.7.6.5.4.3.2.1.3.2.1}}{{10.9.8.7.6.5.4.3.2.1}} = \dfrac{1}{{15}}$
Hence, the required probability $= \dfrac{1}{{15}}$
Thus, the correct option is B.

Note: We need to know the meaning of the factorial for solving problems like that.
(i)Factorial of any positive integer is defined as the multiplication of all the positive integers less than or equal to the given positive integers.
(ii)Factorial of zero is one.
(iii)Factorials are commonly used in permutations and combinations problems.
(iv)Factorials of negative integers are not defined