To solve this problem, we need to find how many two-digit numbers can be formed using the integers 4, 7, 9, 8, and 5 without repetition.

Step-by-step breakdown:

1. A two-digit number consists of two places: the tens place and the units place.
2. For the tens place, we have 5 available digits (4, 7, 9, 8, 5) to choose from.
3. After choosing a digit for the tens place, we cannot repeat it in the units place. This means we have 4 remaining digits to choose from for the units place.

Calculation:

• For the tens place, we have 5 choices.
• For the units place, we have 4 choices (since repetition is not allowed).

Thus, the total number of two-digit numbers is: $5 \times 4 = 20$

Conclusion:

The total number of two-digit numbers that can be formed using the digits 4, 7, 9, 8, and 5 without repetition is 20.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How many three-digit numbers can be formed using the same digits without repetition?
2. What if repetition is allowed? How many two-digit numbers can be formed then?
3. How many odd two-digit numbers can be formed without repetition using these digits?
4. What is the largest two-digit number that can be formed using these digits without repetition?
5. How many two-digit numbers can be formed if the first digit cannot be 4?

Tip: For combinatorics problems, think about the number of choices available at each step and whether repetition is allowed or not!