Digits that strictly increase
How many positive integers have decimal digits that strictly increase from left to right? For example 7, 25 and 1389 count, but 112 and 31 do not.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Can the digit 0 appear?
A number with strictly increasing digits is determined by which digits it uses.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Write the chosen digits in increasing order: each non-empty set of digits gives exactly one such number, and every such number comes from its set of digits.
Step 1: no zero. A 0 could only be the first digit, which is not allowed for a positive integer (and a single 0 is not positive). So the digits come from .
Step 2: sets and numbers. Every non-empty subset of , written in increasing order, gives one number with strictly increasing digits, and different subsets give different numbers.
Step 3. There are non-empty subsets, so 511 such numbers. The largest is 123456789.