Question 1160657
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I tried a few ways to analyze the problem using combinatorics and counting techniques, but it always got very messy.  So I decided the only way I was going to find the answer was by enumerating all the possibilities.<br>
Note that there might be an easy combinatorical path to the answer; but I didn't see it.<br>
For ease of analysis, instead of considering arrangements of ABCDEF with no two consecutive letters next to each other, I modeled the problem as arrangements of 123456 with no two consecutive digits next to each other.<br>
I will describe a bit of the analysis I did and let you have the pleasure of doing the same kind of logical analysis to finish the problem.<br>
Suppose, for example, that the numbers 2 and 4 are first (i.e., Billy and Dahlia are in the first two seats).  Numbers 1, 3, 5, and 6 are left; but the next number can't be either 3 or 5.  So if 2 and 4 are the first two, the first three are either 2-4-1 or 2-4-6.<br>
If the first three are 2-4-1, then 3, 5, and 6 are left.  The 5 and 6 have to be separated, so we have two allowable seating arrangements: 2-4-1-5-3-6 and 2-4-1-6-3-5.<vr>
If the first three are 2-4-6, then 1, 3, and 5 are left.  The 5 can't be next; either the 1 or the 3 can be next, and then the others can be in either order.  This case gives us 4 allowable seating arrangements: 2-4-6-1-3-5, 2-4-6-1-5-3, 2-4-6-3-1-5, and 2-4-6-3-5-1.<br>
Perform a similar analysis for all possible pairs of occupants of the first two seats.<br>
As careful as I tried to be, I found it aggravatingly easy to overlook some allowable arrangements.  As a check on the final list you end up with, note that, for any allowable arrangement, the mirror image arrangement is also allowable -- so the allowable arrangements will occur in pairs.  For example, the arrangements 2-4-1-5-3-6 and 2-4-1-6-3-5 will be matched with the arrangements 6-3-5-1-4-2 and 5-3-6-1-4-2.<br>
I found a total of 90 allowable arrangements....<br>
Have fun....!<br>