1. **Problem statement:** Find the number of 5-digit odd numbers that can be formed with no repeated digits. 2. **Key points:** - A 5-digit number ranges from 10000 to 99999. - The number must be odd, so the last digit must be one of the odd digits: 1, 3, 5, 7, 9. - No digit can be repeated. 3. **Step-by-step solution:** - Step 1: Choose the last digit (units place) which must be odd. There are 5 choices: $\{1,3,5,7,9\}$. - Step 2: Choose the first digit (ten-thousands place). It cannot be zero and cannot be the digit chosen for the last place. - Since the last digit is fixed, we have 9 remaining digits (1-9 except the last digit) for the first digit. - Step 3: Choose the second digit (thousands place). It can be any digit except the two already chosen (first and last digits), including zero. - So, 8 choices remain. - Step 4: Choose the third digit (hundreds place). It can be any digit except the three already chosen. - So, 7 choices remain. - Step 5: Choose the fourth digit (tens place). It can be any digit except the four already chosen. - So, 6 choices remain. 4. **Calculate total number of such numbers:** $$5 \times 9 \times 8 \times 7 \times 6 = 15120$$ 5. **Answer:** There are **15120** five-digit odd numbers with no repeated digits.