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

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.

2024 AMC 10B Problem 20

Answer: A

Difficulty rating: 2080

Solution:

Problem 20 in Other Years