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## Homework Statement

Imagine that there are two lines of people standing by two different cashier spots (cashier 1 and cashier 2) in a department store. A_{1} stands first in line by cashier #1,and is followed accordingly by, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}. In the same manner B_{1} stands first in line by cashier #2, and is followed accordingly by, B_{2}, B_{3}, B_{4}, and B_{5}. Suddenly both Cashier #1, Cashier #2 close their cashier spots and hence, a new line has to be formed all the people in both lines by a completely new Cashier spot, under the condition that the order, A_{1}..A_{6} is maintained just as the order B_{1}..B_{5} is maintained. Furthermore, will A_{3} and B_{2} whom haven't met each other for a long time (and need to talk), stand next to each other in the new formed line. In how many ways may the new line look like?

## Homework Equations

I have tried to understand this problem, how can they stand next to one another if the order has to be maintained. Can somebody shed some light upon this problem...please show all steps... there must be a more efficient way then the way I present below :(

## The Attempt at a Solution

If one considers different orders, you can end up with SO many different possibilities... () marks B2 and A3 they need to be next to each other

Option 1: A1 B1 A2 (B2 A3) B3 A4 B4 A5 B5 A6

Option 2: A1 A2 B1 (B2 A3) B3 B4 B5 A4 A5 A6

Option 3: A1 A2 B1 (B2 A3) A4 A5 A6 B3 B4 B5

Option 4: A1 A2 B1 (B2 A3) B3 A4 B4 B5 A5 A6

Option 5: A1 A2 B1 (B2 A3) B3 A4 A5 B4 B5 A6

Option 6: A1 A2 B1 (B2 A3) B3 A4 A5 A6 B4 B5

Option 7: A1 A2 B1 (B2 A3) B3 B4 A4 A5 A6 B5

Option 8: A1 A2 B1 (B2 A3) B3 B4 A4 B5 A5 A6

Option 9: A1 B1 A2 (A3 B2) B3 A4 B4 A5 B5 A6

Option 10: A1 A2 B1 (A3 B2) B3 B4 B5 A4 A5 A6

Option 11: A1 A2 B1 (A3 B2) A4 A5 A6 B3 B4 B5

Option 12: A1 A2 B1 (A3 B2) B3 A4 B4 B5 A5 A6

Option 13: A1 A2 B1 (A3 B2) B3 A4 A5 B4 B5 A6

Option 14: A1 A2 B1 (A3 B2) B3 A4 A5 A6 B4 B5

Option 15: A1 A2 B1 (A3 B2) B3 B4 A4 A5 A6 B5

Option 16: A1 A2 B1 (A3 B2) B3 B4 A4 B5 A5 A6

...GAH

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