How many different numbers can be formed from the digits 3, 4, 5, 6 and 7 when repetitions are allowed ?

The number of different numbers that can be formed from the digits 3, 4, 5, 6, and 7 when repetitions are allowed is calculated as follows:

Step 1: Identify the number of digits available. In this case, we have 5 digits (3, 4, 5, 6, and 7).

Step 2: Since repetitions are allowed, each place in the number can be filled by any of the 5 digits.

Step 3: If we are forming a one-digit number, there are 5 possibilities (one for each digit).

Step 4: If we are forming a two-digit number, there are 5 possibilities for the first digit and 5 possibilities for the second digit. So, there are 5*5 = 25 possible two-digit numbers.

Step 5: Similarly, for a three-digit number, there are 555 = 125 possible numbers.

Step 6: Continuing this pattern, for a four-digit number, there are 5555 = 625 possible numbers, and for a five-digit number, there are 5555*5 = 3125 possible numbers.

Step 7: To find the total number of different numbers that can be formed, add the number of one-digit, two-digit, three-digit, four-digit, and five-digit numbers.

So, the total number of different numbers that can be formed is 5 + 25 + 125 + 625 + 3125 = 3905.