Question

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba.$

$0$

$1$

$2$

$3$

$4$

Solution:

The initial step would be to choose a letter that could work and build from there.

For instance, if we begin with $a,$ the only letters that can be placed next to it are $c$ or $d.$

From here, we cannot proceed further since the combinations $acbd$ and $acdb$ are not allowed due to the presence of $cb$ and $cd,$ respectively.

The same issue arises if we start with $d,$ as a $b$ would have to be placed in the middle and end up being adjacent to either an $a$ or a $c.$

On the other hand, if we start with a $b,$ the next letter would have to be $d,$ and after that, an $a$ and a $c$ can be placed, making this arrangement possible.

The same methodology holds true for starting with a $c.$ This gives us a total of $2$ rearrangements.

Thus, C is the correct answer.

2015 AMC 10A Problem 10

Solution:

Problem 10 in Other Years