To determine the number of 5-digit decreasing numbers, we can follow a systematic approach:
1. Understanding the Criteria:
- A decreasing number has each digit smaller than the digit to its left. For example, in 87420, each digit is smaller than the one to its left, making it a decreasing number.
2. Strategy:
- To form a 5-digit decreasing number, we need to select 5 digits where each successive digit is smaller than the previous one.
3. Step-by-Step Solution:
- We can start by choosing the last digit first. Since it needs to be the smallest, we have 10 choices (0-9).
- Moving to the next digit to the left, we have one less option, as it must be smaller than the last digit. So, we have 9 choices for the second digit.
- Continuing this pattern, for the third digit, we have 8 choices, for the fourth digit, we have 7 choices, and for the first (leftmost) digit, we have 6 choices.
4. Calculating the Total Number of 5-Digit Decreasing Numbers:
- To find the total number of 5-digit decreasing numbers, we multiply the number of choices for each digit: 10 x 9 x 8 x 7 x 6.
5. Calculating the Result:
- \(10 \times 9 \times 8 \times 7 \times 6 = 30240\)
Therefore, there are 30,240 different 5-digit decreasing numbers.
