csperber wrote:

If we have 7 people in a movie theater, and 3 people choose not to sit next to each other, how many arrangements do we have?

Total ways of seating 7 people in a row = 7 !

if the three sit together,

consider these three as a single unit , we will now have 5 people...these five people can sit in 5! ways...also the three people that are together can themselves sit in 3! ways...

thats a total of 5! * 3!

total ways = 7! - 5! * 3!