Round table. What's the probability that boy/girl alternate. Ten boys and ten girls are sitting at a round table. What is the probability that children would alternate: boy, girl, boy, girl?
Solution:
It's clear that children can be arranged in 19! ways. Now count the ways to place children alternately. There are $\frac{10!^2}{20}$ to do it. I divide by 20 due to circular symmetry. So, the answer is $\frac{10!^2}{20!}$.
Am i right with my solution?
 A: Almost.  Your only error was in estimating there where $10!$ ways to seat boys alternately and $10!$ to seat the girls.  The first boy (or girl) has $20$, not $10$ choices of seats so it is actually $20*9!10!$ rather than $10!10!$.
If you are doing probability it doesn't matter if you take circular symmetry into account or not as long as you do it consistently.
Given seats $1-20$ there are $20!$ ways to place the the kids and to place them alternately there are $20$ places to put the first boy, and $9$ for the second an so on.  There are $20*9!10!$ ways to place to place alternating  boys and girls so the prob is $\frac {20*9!10!}{20!}$.
That's without taking symmetry into account.  If you take symmetry into account.  there are $19!$ ways to do the kids.  And $9!$ ways to place the other $9$ children of the same gender as the child in the top position alternately and $10!$ ways to place the other gender.   So the prob is $\frac {9!10!}{19!}$.
Also if you ignore symmetry and the individual placing of individual children.  There are ${20\choose  10} = \frac {20!}{10!10!}$ ways to place the ten boys and $2$ of them are alternating.  SO the probability is $\frac 2{\frac {20!}{10!10!}}$
And you take symmetry into account by not the idiividual ways of placing children, There are ${19\choose 9}$ ways to choose places for the remaining $9$ of the top gender, or equally ${19\choose 10}$ ways to choose places for the other gender.  And only one of those is alternating.  SO the probability is either $\frac 1{19\choose 9} = \frac 1{\frac{19!}{9!10!}}$ or $\frac 1{19\choose 10} = \frac 1{\frac {19!}{10!9!}}$.
Hopefully it is clear that all of those yield the same answer.
