How many ways to arrange the seating? 
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from ￼ to ￼ in clockwise order. Committee rules state that a Martian must occupy chair ￼1 and an Earthling must occupy chair 15￼ . Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $N(5!)^3$￼ . Find $N$ ￼ .

Please only small hints!
I was starting with a recursion then I saw that there is a limit to how many martians, venusians, earthlings we are allowed to use. 
I was thinking since we can use the formula $(k-1)!/2$ we could so something here. 
Since two are already gone, we have: $(13 - 1)!/2 = 12!/2$ but this doesnt seem correct. 
If we write out the sequence like this:
$M$_ _ _ _ _ _ _ _ _ _ _ _ _ $E$
After $M$ -> $M$ or $E$
After $E$ -> $E$ or $V$
After $V$ -> $V$ or $M$.
Suppose we had:
$M M M M M$ we have $\binom{5}{5}$ ways to place $E$'so:
$M E M E M E M E$ leaving out the first $M$ and the last $E$ there are: $\binom{6}{3} = 20$ ways to arrange the pattern.
But then there is an issue with the $V$'s. mmmmm.
I was thinking casework, but I have nothing to casework on!
Please only small hints!
 A: Some hints:
Consider the directed graph $\Gamma$ with vertex set $\{M, V,E\}$, edges $M\to V$, $V\to E$, $E\to M$, and a loop at each vertex. Any allowed seating pattern $P$ corresponds to a closed path of length $15$ in $\Gamma$ beginning at $M$ and ending at $M$ immediately after an $E$. The allowed patterns can be classified according to how often this path "goes around the center" of the triangle $\{M, V,E\}$.
Call a subword of the form
$$M^\alpha V^\beta E^\gamma, \qquad \alpha\geq1,\  \beta\geq1, \ \gamma\geq1,$$
a circuit. Any allowed pattern $P\in\{M, V, E\}^{15}$ is a concatenation of one to five such circuits.
As an example, let's look at the case of two circuits:
$$P=M^{\alpha_1} V^{\beta_1} E^{\gamma_1}\>M^{\alpha_2} V^{\beta_2} E^{\gamma_2}\ .$$
We have to count in how many ways we can fulfill the conditions
$$\alpha_1+\alpha_2=5, \qquad\beta_1+\beta_2=5, ,\qquad \gamma_1+\gamma_2=5$$
independently of each other. 
A: In case your wondering why it is in the format $N(5!)^3$ is and what $N$ exactly is:

 Given a pattern you can arrange people within race in $5!$ ways so all races in $(5!)^3$ ways. So $N$ is the number of patterns you can construct.

What you now want to know is how many patterns there are. I actually don't know this. my first thought was incorrectly:

 Since you can create a pattern by making a decision between 2 races at each _ it is $2^{13}$. But this is not correct because if you already have MMMM__________M for example you no longer have 2 options at each _

