7people, a, b, c, d, e, f and h, go to a movie and sit next to each other in 8 adjacent seats in the front row of the theatre. in how many different arrangements will there be at least one person between a and f?

There are 30240 ways the 7 people can be arranged

Step-by-step explanation:

When calculating the "at least" value, this means we are looking for the total number of arrangements minus the number of arrangements where A and F are seated together.

To calculate the total number of ways, we use factorial (!)

Total arrangements = 8 x 7 x 6 x 5 x 4 x 3 x 2 = 40320

Together arrangements = 7 x 6 x 5 x 4 x 3 x 2 = 5040  (But A and F can swap seats, so the actual value is double what we calculated)...

5040 x 2 = 10080

Therefore the at least value can be calculated:

40320 - 10080 = 30240