in how many ways A B C D E F are arranged so that A and E always side by side or never side by side.
Answers
Answer:
Number of ways of arranging A, B, C, D, E, F with A and E together is 240.
Number of ways with A and E not together is 480.
Step-by-step explanation:
The total number of ways of arranging the 6 letters is 6! ( note that 6! = "6 factorial" = 6×5×4×3×2×1 ).
Arranging the letters so that A and E are together amounts to arranging the 5 objects B, C, D, F, AE, and there are 5! = 120 ways of doing this. Since the "AE" object can actually be either AE or EA, we need to double the count.
So the total number of ways of arranging A, B, C, D, E, F with A and E together is 2 × 5! = 240.
The arrangements of A, B, C, D, E, F with A and E not together is then just all the "other" arrangments, so their number is
6! - 2 × 5! = 6 × 5! - 2 × 5! = 4 × 5! = 480.