— Burt Kaliski Jr. (@modulomathy) May 2, 2012
We’ll start by assuming that single-digit numbers have their single digit in increasing order. 9 numbers so far.
Among the two-digit numbers, 36 are in increasing order: 8 in the 10s, 7 in the 20s, …, up to 1 in the 80s; 8+7+…+1 = 36.
The 100s and 200s are similar to the two-digit numbers, except that we lose the 10s in the first case and the 10s and 20s in the second. Another 28 + 21 numbers.
In the 300s, we have just the 5 in the 40s and the 4 in the 50s. None in the 60s, whether a leap year or not; 367 would be the first one.
Total: 9 + 36 + 28 + 21 + 5 + 4 = 103.