2015 AMC 10A Problem 10

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

ba.

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Solution

The forbidden adjacent pairs are

ab,ba,bc,cb,cd,dc.

If an arrangement starts with

a,

its second letter must be

c

d.

After

ac,

either remaining order contains

cb

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.