How many positive integers less than 600 can be formed using the numbers 1,3,4…

Concept

The number of ways to arrange r items chosen from n distinct items without repetition is the permutation count nPr = n × (n-1) × ... × (n-r+1); when a number is built digit by digit without repeating any digit, the same falling-product rule applies to each place value, and an upper-bound condition on the number simply restricts how many digits are valid for the leading (highest) place.

Application

1. One-digit numbers: each of the 4 given digits (1, 3, 4, 6) is itself a valid number less than 600, so there are 4 one-digit numbers.

2. Two-digit numbers: the first digit can be any of the 4 digits and the second digit any of the remaining 3 (no repetition), giving 4 × 3 = 12 arrangements. Every two-digit number formed from {1, 3, 4, 6} is automatically below 600, so all 12 count.

3. Three-digit numbers under 600: a three-digit number starting with 6 is at least 600, so the hundreds digit must be 1, 3, or 4 — 3 valid choices. Once the hundreds digit is fixed, the tens digit can be any of the remaining 3 digits, and the units digit any of the remaining 2, giving 3 × 2 = 6 arrangements per hundreds-digit choice. Total three-digit numbers = 3 × 3 × 2 = 18.

4. Adding the three cases: 4 + 12 + 18 = 34 positive integers in all.

Cross-check

An independent check: ignoring the upper-bound restriction, all three-digit arrangements of the 4 digits number 4 × 3 × 2 = 24; exactly those starting with 6 (6 _ _) must be excluded, and there are 1 × 3 × 2 = 6 such arrangements, leaving 24 − 6 = 18 — matching the count above and confirming the total of 34.