Number of Integers Greater Than 6000 Formed Using Digits 3, 5, 6, 7, and 8 without Repetition

To find the number of integers greater than 6000 that can be formed using the digits 3, 5, 6, 7, and 8 without repetition, we need to consider the possible positions of each digit in the number.

1. Determine the First Digit:
To form a number greater than 6000, the first digit must be either 6, 7, or 8. Therefore, we have 3 options for the first digit.

2. Determine the Second Digit:
Since we cannot repeat any of the digits, the second digit can be any of the remaining 4 digits (excluding the first digit). Therefore, we have 4 options for the second digit.

3. Determine the Third Digit:
Similarly, the third digit can be any of the remaining 3 digits. Thus, we have 3 options for the third digit.

4. Determine the Fourth Digit:
The fourth digit can be any of the remaining 2 digits. Hence, we have 2 options for the fourth digit.

5. Determine the Fifth Digit:
Finally, the fifth digit can only be the remaining 1 digit.

Calculating the Total Number of Integers:
To calculate the total number of integers greater than 6000 that can be formed without repetition, we multiply the number of options for each digit.

Number of options for the first digit = 3
Number of options for the second digit = 4
Number of options for the third digit = 3
Number of options for the fourth digit = 2
Number of options for the fifth digit = 1

Total number of integers = 3 × 4 × 3 × 2 × 1 = 72

Therefore, there are a total of 72 integers greater than 6000 that can be formed using the digits 3, 5, 6, 7, and 8 without repetition.