Question

Six cards numbered $1$ through $6$ are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.

答案

Solution:

First count arrangements from which some card's removal leaves the rest ascending. Any such arrangement arises by choosing the card to remove ( $6$ ways) and inserting it into one of the $6$ gaps of the other five cards written in increasing order, for $36$ constructions. But the fully sorted row $123456$ arises from all $6$ card choices, and each of the $5$ arrangements obtained by swapping two adjacent cards of the sorted row arises twice (move either card of the pair past the other). Every other construction gives a distinct arrangement.

So the ascending count is $1 + 5 + (36 - 6 - 10) = 26,$ and by symmetry there are $26$ descending arrangements. No arrangement is counted in both totals: that would require an ascending and a descending subsequence of five cards, needing at least $5 + 5 - 1 = 9$ cards.

The total is $26 + 26 = 52.$

2020 AIME I Problem 5

Solution:

Problem 5 in Other Years