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?

$60$

$72$

$90$

$108$

$120$

Solution:

Scan the row: wherever a left shoe touches a right shoe, they have to be mates. Look at the pattern of sides (L or R) across the six spots; every switch between L and R must sit at a matched pair. Two patterns keep all lefts together then all rights, $LLLRRR$ and $RRRLLL.$ Each has a single switch, so pick the mated pair there ( $3$ ways) and order the other two lefts ( $2$ ) and two rights ( $2$ ): $12$ each. The other allowed patterns $LLRRRL, LRRLLR, LRRRLL, RLLLRR, RLLRRL, RRLLLR$ give $6$ arrangements apiece. Altogether $2 \cdot 12 + 6 \cdot 6 = 60.$ Therefore, the answer is A.

2024 AMC 10B Problem 20

Solution:

Problem 20 in Other Years