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In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?

 May 4, 2024
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There are two main cases to consider for how the siblings can be arranged:

 

Case 1: Two pairs sit together in one row

 

There are 3 ways to choose which row will have two pairs of siblings sitting together. For the chosen row, there are 2 ways to arrange the two siblings within each pair (brother first or sister first). Once we've placed these four children, the remaining two children can fill the remaining seats in the other row in 2!=2 ways. This gives a total of 3⋅2⋅2=12​ arrangements for this case.

 

Case 2: All pairs are split between the rows

 

Here, there is only one way to arrange the siblings: each sibling sits in a different row from their brother or sister. Once we've placed one child from each pair in the first row, the remaining three children can fill the second row in 3!=6 ways. So, there are 1⋅6=6​ arrangements for this case.

 

Adding the number of arrangements for these two cases, there is a total of 12+6=18​ ways to seat the siblings.

 May 4, 2024

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