What theorem or concept should I use to approach this problem>

Question

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?

Answer 1

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.

Answer 2

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:

1. AAAABBBB
2. AAABABBB
3. AAABBABB
4. AAABBBAB
5. AABAABBB
6. AABABABB
7. AABABBAB
8. AABBAABB
9. AABBABAB
10. ABAAABBB
11. ABAABABB
12. ABAABBAB
13. ABABAABB
14. ABABABAB

More generally, if the counts of the two objects can be different, this is the ballot problem.