Introduction:
To find the number of 4-digit numbers greater than 5000 that can be formed using the digits 3, 4, 5, 6, and 7 (without repetition), we need to consider the possible combinations of these digits in the thousands, hundreds, tens, and units place.

Step 1: Determine the possibilities for the thousands place:
Since we are looking for numbers greater than 5000, the thousands place can only be filled with the digits 5, 6, or 7. Therefore, there are 3 possibilities for the thousands place.

Step 2: Determine the possibilities for the hundreds, tens, and units place:
For the hundreds, tens, and units place, we can use any of the remaining digits (3, 4, 5, 6, or 7) as long as they are not repeated.

- For the hundreds place, there are 4 remaining digits to choose from.
- For the tens place, there are 3 remaining digits to choose from.
- For the units place, there are 2 remaining digits to choose from.

Therefore, the total number of possibilities for the hundreds, tens, and units place is 4 x 3 x 2 = 24.

Step 3: Calculate the total number of possibilities:
To find the total number of 4-digit numbers greater than 5000, we multiply the possibilities for the thousands place (3) by the possibilities for the hundreds, tens, and units place (24).

Total number of possibilities = 3 x 24 = 72.

Conclusion:
There are 72 different 4-digit numbers greater than 5000 that can be formed using the digits 3, 4, 5, 6, and 7 without repetition.

3P1×4P3