How many positive integers less than 500 can be formed using the digits 1, 2,…

Concept: When forming numbers from a fixed set of digits with each digit used at most once, split the count by how many digits the number has, then apply the multiplication principle for each digit position — the number of choices for a position is the digits still unused, so it decreases by one at every subsequent position.

1. One-digit numbers: any of the 4 digits (1, 2, 3, 5) gives a valid one-digit number, and every one-digit number is automatically less than 500. Count = 4.

2. Two-digit numbers: the tens digit can be any of the 4 digits and the units digit any of the remaining 3 (no repetition), giving 4 × 3 = 12 numbers. The largest possible two-digit number here is 53, so all 12 are already less than 500.

3. Three-digit numbers: to stay below 500 the hundreds digit must come from {1, 2, 3} — placing 5 in the hundreds position would make the number 500 or more — so the hundreds digit has 3 choices; the tens digit then has 3 remaining choices (any digit except the one already used); the units digit has 2 remaining choices. Count = 3 × 3 × 2 = 18.

Cross-check: without the less-than-500 restriction, the number of 3-digit numbers formable from the 4 digits is 4 × 3 × 2 = 24. Exactly those starting with 5 must be excluded — with 5 fixed as the hundreds digit, the remaining two positions can be filled in 3 × 2 = 6 ways. So the restricted count is 24 − 6 = 18, confirming the value above.

Total = 4 + 12 + 18 = 34.