What theorem or concept should I use to approach this problem> I have 4 objects in group A and 4 objects in group B. 
One by one I need to add the 8 objects from the two different groups in a line. 
But the condition is that as the line is being created, the amount of "A" objects is always >= the amount of "B" objects. 
What I have 4 objects in group A and 4 objects in group B. 
One by one I need to add the 8 objects from the two different groups in a line. 
But the condition is that as the line is being created, the amount of "A" objects is always >= the amount of "B" objects. 
I need to count the total number of possible arrangements
What theorem or concept should I use?  or concept should I use? 
 A: This is one of the many things counted by the Catalan numbers $\frac{1}{n+1}\binom{2n}{n}$.  In particular, $n=4$ yields $\frac{1}{5}\binom{8}{4}=14$. Explicitly, the arrangements are:


*

*AAAABBBB

*AAABABBB

*AAABBABB

*AAABBBAB

*AABAABBB

*AABABABB

*AABABBAB

*AABBAABB 

*AABBABAB

*ABAAABBB

*ABAABABB

*ABAABBAB

*ABABAABB

*ABABABAB


More generally, if the counts of the two objects can be different, this is the ballot problem.
A: This can also be posed as counting the number of unilateral walks of type $n$; that is, strings of $n$ '$+1$'s and $n$ '$-1$'s whose partial sum is never negative.  Let $w(n)$ be the number of such walks. Furthermore, let $2k$ be the number of steps at which the walk first returns to the starting point. Such a walk looks like
$$
\left<+1\right>\left<\text{walk of type $k-1$}\right>\left<-1\right>\left<\text{walk of type $n-k$}\right>
$$
Thus, we get the recurrence
$$
w(n)=\sum_{k=1}^nw(k-1)w(n-k)
$$
In the answer cited, Generating Functions are used to compute
$$
w(n)=\frac1{n+1}\binom{2n}{n}
$$
which is the $n^\text{th}$ Catalan number.
