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? 

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.
