Question

$12$ boys and $2$ girls are to be seated in a row such that there are at least $3$ boys between the $2$ girls. The number of ways this can be done is $\lambda$ 12!, then the value of $\lambda$ is
$\left( a \right){\text{ 55}}$
$\left( b \right){\text{ 110}}$
$\left( c \right){\text{ 20}}$
$\left( d \right){\text{ 45}}$

Answer 1

First of all we will calculate the arrangement for the two girls when they are together and then will calculate the one boy who sits between the two girls and also the two boys sit between the two girls. And after all, we will subtract this from the total arrangement which will be $14!$.

Formula used:
Permutation,
$^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$

Complete step-by-step answer:
So the arrangement for the two girls who are together will be
$\Rightarrow 2 \times 13 = 26$
Now the one boy who sits between the two girls will be calculated as
$\Rightarrow 2 \times 12 = 24$
For the two boys who sit between the two girls, the arrangement will be as following
$\Rightarrow 2 \times 11 = 22$
Hence, from the question, we can easily calculate the total arrangement and it will be equal to

Now we have to calculate the required arrangements, so it will be
$\Rightarrow 14! - \left( {26 + 24 + 22} \right)12!$
And on solving it will be equal to
110 x 12!
As we have already given the number of ways it can be done which is
$\lambda \times 12!$
Therefore, on equating, we get
$\Rightarrow 110 \times 12! = \lambda \times 12!$
And on solving, we get
$\Rightarrow \lambda = 110$
Therefore, the option $\left( b \right)$ is correct.

Additional information:
Permutation means all possible arrangements of the number, letter, or any other product or things, etc. Well, the most basic difference in that permutations is ordered sets. The permutation is a course of action of things where the request for a game plan matters. The situation of everything in change matters. Consequently, Permutation can be related to Position.

Note: In Permutation, arrangement matters in Combination, the arrangement does not matter. So while solving this type of problem we should have to keep in mind whether it is of arrangement or it is of grouping. So we should always be kept in mind while solving such types of problems so that we cannot get the error and can solve it easily without any problem.