A group of 30 people consists of 15 men and 15 women. How many ways are there to
(a) arrange all the people people in a row?
30! = 265252859812191058636308480000000
(b) arrange all the people in a row so that the men are together?
The men can be in positions 1 thru 15, 2 thru 16, ..., 16 thru 30
For each of those 16 positions to place the men, we can arrange them in
each position in 15! ways, Then we can arrange the 15 women in 15! ways.
That's 16*15!*15! = 27360196043587190784000000
(c) divide all the people into two groups (group 1 and group 2) so that all the men are in group 1 and all the women are in group 2?
Order within the groups does not matter, so there is just 1 way.
Put all the men in group 1 and all the women in group 2.
Answer: 1 way.
[Are you sure that was how that one was stated? It seems too easy]
Edwin