Combinatorics: Using the Glue Method vs. Slot Method

[NMencia09]

Hello,

This is from the World Translations Strat. Guide. pg. 73 under "Arrangement with Constraints"

Greg, Marcia, Peter, Jan, Bobby, and Cindy go to a movie and sit next to each other in 6 adjacent seats. If Marcia and Jan will not sit next to each other, in how many different arrangements can the six people sit?

How can you solve this problem using the Slot Method?

There are 6 separate decisions, but the choices for the slots depend on who sits down first. If Marcia sits down first, then there I'm getting 6,4,4,3,2,1 for the slots which comes ou to 576. The answer is 480 using glue method. Is there a way to use the S.M.?

Thank you!

[LazyNK]

Hey Noah,
I believe you must get used to the glue method for such problems. The slot method is possible, but very complicated. I am not sure how you arrived at the slot method results you mention, but the correct slot method is as follows ( and I'm sure looking at this solution, you'd see the advantage of the glue method) :
GMPJBC
Case 1 : M takes first slot from left, N takes third from left
Total possibilities = 1x4x1x3x2x1 = 24
Case 2 : M takes first slot from left, N takes fourth from left
Total possibilities=1x4x3x1x2x1=24
Case 3 : M takes first slot from left, N takes fifth from left
Total possibilities=1x4x3x2x1x1=24
Case 4 : M takes first slot from left, N takes sixth slot
Total possibilities=1x4x3x2x1x1=24
Total allowed possibilities where M takes first slot= 24x4 = 24 x # of allowed possibilities for N=96
Similarly we'd have Case5-Case7 where M takes the second slot from left (only three allowed possibilities for N for this case)=24x3=72
Similarly, Case8-Case10 where M takes the third slot from left (again only three allowed possiblities for N)=24x3=72
As again, Case11-Case13, M takes fourth slot from left(three allowed possiblities for N)=24x3=72
Case14-Case16, M takes fifth slot from left (three allowed possiblities for N)=24x3=72
Case17-Case20, M takes sixth slot from left ( Four allowed possiblities for N)=24x4=96

Thus total allowed possibilities=96x2+72*4=192+288=480

Hope it clarifies.
NK

[tim]

Thanks Lazy! Keep in mind not every conceivable strategy can be applied to every conceivable problem..