How many positive integers less than 600 can be formed using the digits 1, 3,…

Concept

When digits cannot repeat within a single number, the Fundamental Counting Principle says the number of ways to fill each place is the count of digits still unused, and these place-counts multiply together for numbers of a given length. A "less than N" restriction is handled by first fixing which digits are allowed to occupy the leading (highest) place, since that alone caps how large the number can get.

Working

1. Numbers of length 1: any of the 4 digits {1, 3, 4, 6} stands alone as a valid number under 600. Ways = 4.

2. Numbers of length 2: every 2-digit arrangement of these digits is under 600 automatically (the largest possible is 64), so the tens place has 4 digit choices and the units place has 3 remaining choices. Ways = 4 x 3 = 12.

3. Numbers of length 3 that must stay under 600: the hundreds place can only take a digit from {1, 3, 4} -- using 6 there would push the number to 600 or above -- giving 3 choices; the tens place then has 3 digits left to choose from, and the units place has 2 left. Ways = 3 x 3 x 2 = 18.

4. Add the three disjoint cases together: 4 + 12 + 18 = 34.

Cross-check

Cross-check: ignoring the "less than 600" restriction, all 1-, 2- and 3-digit arrangements of these 4 digits total 4 + 12 + 24 = 40. The restriction removes exactly the three-digit numbers with 6 in the hundreds place, and there are 1 x 3 x 2 = 6 of those. 40 - 6 = 34, confirming the count.