In how many ways can 8 people be seated in a row of chairs if three of the people, John, Wilma and Paul, refuse to sit in three consecutive seats?
The total arrangements possible = 8!
Now...let's consider the arrangements possible if all three DO sit together [in consecutive seats ]
Considering them as a single entity....they can be arranged as 3! = 6 ways
And they can be seated in any 6 positions [ positions 1 - 6 ] = 6 ways
And for each of these arrangements the other people can be seated in 5! = 120 ways
So...the total arrangements possible if they DON'T sit together =
Possible arrangements - Arrangements where they sit together =
8! - [ 6 * 6 * 5!] =
40320 - 36*120 =
40320 - 4320 =