2015 AMC 10A Problem 10

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

Forbid the adjacent alphabet pairs in either order

Starting with

b

or

c

is the only way to finish

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.

Problem 10 in other years