Number of 4-digit Numbers Greater than 5000

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 can break down the problem into smaller steps.

Step 1: Counting the possibilities for the thousands digit

Since we want the number to be greater than 5000, the thousands digit must be either 5, 6, or 7. Therefore, there are three possibilities for the thousands digit.

Step 2: Counting the possibilities for the hundreds digit

After fixing the thousands digit, we are left with four digits (3, 4, 6, and 7) to choose from for the hundreds digit. Since no digit can be repeated, there are four possibilities for the hundreds digit.

Step 3: Counting the possibilities for the tens digit

Similarly, after fixing both the thousands and hundreds digits, we have three digits (3, 4, and 6) to choose from for the tens digit. Again, there are three possibilities for the tens digit.

Step 4: Counting the possibilities for the units digit

Finally, after fixing all three previous digits, we have two digits (3 and 4) to choose from for the units digit. Hence, there are two possibilities for the units digit.

Step 5: Calculating the total number of possibilities

To find the total number of possibilities, we multiply the number of possibilities at each step:

Number of possibilities for the thousands digit = 3
Number of possibilities for the hundreds digit = 4
Number of possibilities for the tens digit = 3
Number of possibilities for the units digit = 2

Total number of possibilities = 3 x 4 x 3 x 2 = 72

Therefore, 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.