Suppose that there are m girls and n boys in a class. What is the number of ways of arranging the students in a line so that all the girls are together? Explain?

Question

kinggeorge · Answer

Since all the girls have to be together, we can consider them as a single girl (at first). Thus, you have a line of n boys, and 1 girl. This means that there are $n+1$ places for the girl to stand. 

Now we break up the single girl into the line of m girls again. We need to find the total number of possible orderings of this. This is $m!$. 

Hence, the number of ways of arranging the students in a line so that all the girls are together is given by $$m! (n+1)$$

kinggeorge · Answer

Forgot one thing. Give me a second.

anonymous · Answer

Okay... Thanks!

kinggeorge · Answer

We also have to consider the possible orderings of the n boys. Thus, we also have to multiply by $n!$. 

Final answer should then be $$m!n!(n+1)$$

anonymous · Answer

Thank You so much
!

kinggeorge · Answer

You're welcome.