2017 AIME II Problems/Problem 4
Find the number of positive integers less than or equal to whose base-three representation contains no digit equal to .
The base- representation of is . Because any -digit base- number that starts with and has no digit equal to must be greater than , all -digit numbers that have no digit equal to must start with or in base . Of the base- numbers that have no digit equal to , there are -digit numbers that start with , -digit numbers that start with , -digit numbers, -digit numbers, -digit numbers, -digit numbers, -digit numbers, and -digit numbers. Summing these up, we find that the answer is .
Note that , and . There can be a digit number less than , and each digit can either be or . So one digit numbers and so on up to digit.
Now we have to subtract out numbers from to
Then either the number must begin or with four more digits at the end
Using s and s there are options for each so:
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