# Four Children Combination Problem

This is a problem that has haunted me for some time

There is two family. First Family :

  Husband : Adam
Wife : Barbie
Son1 : Caran
Son2 : Dave
Daughter1 : Emily
Daughter2 : Fiona


Second Family :

  Husband : George
Wife : Honey
Son1 : Ian
Son2 : Jack
Daughter1 : Kate
Daughter2 : Laura


One day, Adam is in love with Honey and George is in love with Barbie. Now, everyone decide to divorce his/her partner. Then, Barbie married to George and Adam married to Honey. Both new couple decided to keep 4 children having two girl and two boy. And, Every parent keep his/her own two children.

Question

What is the combination for 4 children having two boy and two girl and two of childrens are from one of his/her parent?

UPDATE

It may be two boy from one parent, two girls from another. Or It may be 1 boy and 1 girl from one parent, one boy and one girl from his/her partner.

-

Barbie/George/Dave/Fiona/Ian/Laura

Each parent keeps two of their own kids (one of each sex). Is that the question?

EDIT: I will just answer it.. there are $4~ choose~ 2 = 6$ options to pick kids for the first family. Two of those options result in picking two kids of the same sex and the other four correspond to mixed sex choices.

For a mixed sex choice there are $(4~ choose~ 2) -2 = 4$ choices for picking kids from the second family to join the first (the minus 2 comes from the fact that I cannot choose two same sex kids and there are two options corresponding to this case).

For a same sex choice there is only one option for picking kids from the second family.

So, how many choices of family configurations are there? Just add them up.. 4+4+4+4+1+1 = 18

-
It may be two boy from one parent, two girls from another. Or It may be 1 boy and 1 girl from one parent, one boy and one girl from his/her partner. –  Shashi Jan 14 '14 at 11:09
This simplifies things actually. So without even doing anything technical you can just enumerate the numbers (if the same sex is allowed it just adds to the total but not much). –  John U Jan 14 '14 at 11:16
Yes... And Total will be 4 with combination of two boy and two girl... –  Shashi Jan 14 '14 at 11:21
Improved hint: Start by "choosing" two of your own four kids. There are 6 different combinations here. For a particular "choice" of your own kids, now how many choices do I have to pick from on the other side (it is dependent on the first choice but it is reasonably straightforward to see the pattern)? –  John U Jan 14 '14 at 11:27
You make it very much easy... It is very much easy... –  Shashi Jan 14 '14 at 11:47