2024 AMC 10B Problem 20

Question

Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?

60

72

90

108

120

Answer: A

Concepts:arrangements with restrictions casework

Difficulty rating: 2080

Small hint:

The rule says: wherever a left shoe touches a right shoe, they must be mates from the same pair

Big hint:

Case on the left/right pattern of the six spots; each place where the side switches pins down one pair

Solution:

Wherever an

L

meets an

R

in the side pattern, those two shoes must be mates. Thus an interior run cannot have length

1:

its lone shoe would have to be the mate of both neighbors. With three

L

’s and three

R

’s, the only possible patterns are

LLLRRR,

RRRLLL,

LLRRRL,

LRRLLR,

LRRRLL,

RLLLRR,

RLLRRL,

and

RRLLLR.

For either one-switch pattern, choose the pair at the switch and order the remaining shoes, giving

3\cdot2\cdot2=12

arrangements. Each of the other six patterns has

6

assignments of the three pairs to its switches. Hence the total is

2\cdot12+6\cdot6=60.

Therefore, the answer is A.