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 $ba.$

$0$

$1$

$2$

$3$

$4$

Answer: C

Concepts:arrangements with restrictions casework

Difficulty rating: 1370

Solution:

The forbidden adjacent pairs are $ab,ba,bc,cb,cd,dc.$

If an arrangement starts with $a,$ its second letter must be $c$ or $d.$

After $ac,$ either remaining order contains $cb$ or $cd.$ After $ad,$ the remaining $b$ and $c$ must be adjacent. Thus no valid arrangement starts with $a.$

By symmetry, no valid arrangement starts with $d.$

If an arrangement starts with $b,$ it must continue with $d,$ then $a,$ then $c,$ giving $bdac.$

Similarly, starting with $c$ gives only $cadb.$ Therefore there are $2$ valid rearrangements.

Thus, C is the correct answer.

2015 AMC 10A Problem 10

Answer: C

Difficulty rating: 1370

Solution:

Problem 10 in Other Years