Take the task of arranging the 6 students in a row and break it into stages.
NOTE: We're going to ignore the chairs and just examine how many ways there are to arrange the 6 children in a row.
The answer will be the same.

Stage 1: Arrange Arya, Chen, Daniel and Franco is a row
We can arrange n unique "objects" in a row in n! ways
So, we can arrange these 4 children in 4! ways (24 ways)
So, we can complete stage 1 in 24 ways

IMPORTANT STEP: For each of the 24 arrangements of Arya, Chen, Daniel and Franco, add a space on either side of each child.
For example, one of the possible arrangements is Chen, Daniel, Franco, Arya
So, add spaces as follows: ___Chen___Daniel___Franco___Arya___
We'll now place Betsy in a space and place Emily in a different space.
This will ENSURE that Betsy and Emily are not seated together.

Stage 2: Choose a space to place Betsy
There are 5 spaces to choose from.
So we can complete stage 2 in 5 ways

Stage 3: Choose a space to place Emily
Once we select a space in stage 2, there are 4 spaces remaining to choose from.
So we can complete stage 3 in 4 ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all 6 children) in (24)(5)(4) ways (= 480 ways)

Answer: B

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.

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