SOLUTION: 10 people line up for a photo, including you and your friend. How many line arrangements are there if the photographer insists on keeping 3 people in between you and your friend?

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Question 352388: 10 people line up for a photo, including you and your friend. How many line arrangements are there if the photographer insists on keeping 3 people in between you and your friend?
Answer by nyc_function(2741) About Me  (Show Source):
You can put this solution on YOUR website!
There are 10 people: A(you), B(your friend) and 8 others.

You and your friend are spaced like this:
A____B or B____A = two ways

Select 3 of the Others and arrange them in the spaces.

There are: 8P3 = 8!/5! = 336 ways.

Now we have 6 "people" to arrange: AxyzB and 5 others.

There are: 6! = 720 ways.

Therefore, there are: 2 * 336 * 720 = 483,840 line arrangements.