Question

How many ways are there to arrange the letters a, b, c, and d such that a is not followed immediately by b?

Answer 1

First, let's determine how many combinations there are without any restrictions. There are 4 letters, so we have 4 options to choose from for the first letter. For the second letter, there are only 3 options, for the third there are only 2 options, and finally for the last letter there is just 1 option. Therefore, we have:

$4! = 4 \times 3 \times 2 \times 1 = 24$ options

Now we can subtract the number of combinations in which a is immediately followed by b.

Instead of having 4 options to choose from, we now only have 3:

ab, c, or d

So using the same logic as before, we have:

$3! = 3 \times 2 \times 1 = 6$ options

Now just subtract the 6 combinations that don't work from the total:

24 - 6 = 18

The answer is 18.