We can disregard and because they are fixed at the beginning and end, respectively. There are 4! ways to arrange the Bs and Cs without any restrictions; in 1/2 of these comes before , and in half of the ones with before we also have before . So the total number of arrangements is

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I'm not sure what you mean. If you start with a B, do you have to end with a B? And do you have to start with ?(1) What if each time, our trip starts by or , we get a total of ?

No, .(2) What about when there are 8 places, is it ?

Thanks for the help!