A committee of 14 people consists of 4 boys, 5 girls and 5 other people are arranged in a row. Given that the boys are separated, find the probability of the girls are not all together
Of course
other people must be either male or female. If the there are \(\displaystyle m\) total males & \(\displaystyle f\) total females in order for the males to be separated it has to be the case that \(\displaystyle 4\le m\le 7\). Given that the males are separated there must be at least \(\displaystyle m-1\) females.
Could there be a committee of these people in which \(\displaystyle m=8~\&~f=6~?\) Well yes there could because \(\displaystyle m+f=14\) but there would be no way to separate all of the males; there are only six separators that create seven separate spaces.
Now a word about the phrase "
not all together" it means that at least two of the same sex are standing together.
Let's take an example: \(\displaystyle m=6~\&~f=8\). If this example it takes five females to separate the six males
but there are not enough males to separate the females. Anyone tackling this question needs to fully understand the setup.
There are \(\displaystyle 14!= 87,178,291,200 \) ways to arrange fourteen people in a row.
There are \(\displaystyle (8!)\left(^9\mathcal{C}_6\right)(6!)=2438553600\)
ways to arrange the committee so that no two of the six males are together.
Explanation: There are \(\displaystyle 8!\) ways to arrange the females who create nine places to separate the six males who can be arranged in \(\displaystyle 6!\) ways.
Thus if we are given six men and eight women to arrange in a row the probability that all the six men are separated is \(\displaystyle \frac{4}{143}\)
SEE HERE
Now the downer: I have no idea how to work this question. It is so poorly put as to be unintelligible.
1) How is the committee chosen? Are the first 4 boys & 5 girls fixed?
2) Even if they are or are not, how is the makeup of the other five people determined?
3) Was the complete question given?
I hope someone reading this or the author of the O.P. can help us here.