combination and permutation

[TCMM]

need help in understanding this

12a)
since in no order 10men and 10 women
total pax 20
20! ways of arranging these people in a row

**12b) - (need help in understanding this concept, new to maths)

 

[Steven G]

How many ways can you separate the 20 people into two tables with 5 people each? Please do this one and then we'll proceed from there.

 

[pka]

How many ways can you separate the 20 people into two tables with 5 people each? Please do this one and then we'll proceed from there.

I think that someone has misread the question. As I read the question, it has two parts. The b) part says there are two tables, one blue and one red. There are twenty people, ten men and ten women. We are to seat that group of people at the two tables in such a way that each table has five men and five women in alternating seats. Lets count the number of ways to seat five women and five men at the blue table.
There are [imath]\dbinom{10}{5}\cdot\dbinom{10}{5}=63504[/imath] ways to choose five men and five women from the group to be seated in alternating seats at the blue table. The remaining folks can be seated at the red table. From the group of selected men choose the oldest and seat him at the table. Once he is seated, the table is completely ordered. Moving clockwise from his left choose men to be seated in every other seat.
That can be done is [imath]4![/imath] ways. In how many ways can the five ladies be seated? What is the total number of ways to seat the twenty people?