2022 AMC 10 A
Complete problem set with solutions and individual problem pages
Suppose that cards numbered , , ,, are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards , , are picked up on the first pass, and on the second pass, on the third pass , , , on the fourth pass, and , , on the fifth pass. For how many of the ! possible orderings of the cards will the cards be picked up in exactly two passes?
- A.
- B.
- C.
- D.
- E.
Picking up in exactly two passes is equivalent to cutting cards in half.
For example, .
When we rearrange, we need to make sure are arranged in increasing order, but we cannot include the one pass .
:
(Note: there are possible spot for , but we need to subtract the situation when [1] is at the first place.)
:
:
In total: .
