2023 AMC 12A Problems/Problem 24
Problem
Let be the number of sequences , , , such that is a positive integer less than or equal to , each is a subset of , and is a subset of for each between and , inclusive. For example, , , , , is one such sequence, with .What is the remainder when is divided by ?
Solution 1
Consider any sequence with terms. Every 10 number has such choices: never appear, appear the first time in the first spot, appear the first time in the second spot… and appear the first time in the th spot, which means every number has choices to show up in the sequence. Consequently, for each sequence with length , there are possible ways.
Thus, the desired value is
~bluesoul