To determine how many three-digit numbers can be formed from the digits of the number 32564 without repeating any of the digits, we will follow a systematic approach. ### Step-by-Step Explanation 1. **Identify the Digits**: The digits available from the number 32564 are 3, 2, 5, 6, and 4. This gives us a total of 5 unique digits to work with. 2. **Choosing the First Digit**: For a three-digit number, the first digit cannot be zero (but in this case, we don't have zero among our digits). Therefore, we can choose any of the 5 digits (3, 2, 5, 6, or 4) as the first digit. This gives us **5 options** for the first digit. 3. **Choosing the Second Digit**: After selecting the first digit, we cannot use that digit again (since we are not allowed to repeat digits). This leaves us with 4 remaining digits to choose from for the second digit. Thus, we have **4 options** for the second digit. 4. **Choosing the Third Digit**: After selecting the first and second digits, we again cannot use those two digits. This leaves us with 3 remaining digits to choose from for the third digit. Therefore, we have **3 options** for the third digit. 5. **Calculating the Total Combinations**: To find the total number of different three-digit combinations, we multiply the number of choices for each digit together: \[ \text{Total Combinations} = (\text{Choices for First Digit}) \times (\text{Choices for Second Digit}) \times (\text{Choices for Third Digit}) \] \[ \text{Total Combinations} = 5 \times 4 \times 3 = 60 \] ### Conclusion Thus, the total number of three-digit numbers that can be formed from the digits 32564 without repeating any digits is **60**. Therefore, the correct option is **D**. ### Explanation of Other Options - **Option A (120)**: This option suggests that there are more combinations than possible. The calculation shows that we only have 60 combinations, not 120. - **Option B (10)**: This option is too low. It does not account for the multiple choices available for each digit. - **Option C (20)**: This option is also too low. It underestimates the number of combinations by not considering all available digits for each position. ### Revision Summary - We can form three-digit numbers using 5 unique digits (3, 2, 5, 6, 4). - The first digit has 5 options, the second has 4 options, and the third has 3 options. - The total number of combinations is calculated as \(5 \times 4 \times 3 = 60\). - The correct answer is **D (60)**, as it accurately reflects the number of unique three-digit combinations possible.