Question: A small combination lock on a suitcase has 6 wheels, each labeled with the 10 digits 0 to 9. How…

A small combination lock on a suitcase has 6 wheels, each labeled with the 10 digits 0 to 9. How many 6 digit combinations are possible if no digit is repeated? If digits can be repeated? If successive digits must be different? … a. The number of different 6 digit combinations possible if no digit is repeated is

Solution

To find the number of different 6-digit combinations possible if no digit is repeated, follow these steps: First, consider that for a combination with no repeated digits, you have 10 options for the first digit. Then, for the second digit, since it cannot repeat the first, you have 9 options. Next, for the third digit, you have 8 options because it must differ from the first two. Continuing this logic: For the fourth digit, you have 7 options. For the fifth digit, you have 6 options. For the sixth digit, you have 5 options. Multiply these possibilities together to find the total number of combinations: \[ 10 \times 9 \times 8 \times 7 \times 6 \times 5 \] Calculate: \[ 10 \times 9 = 90 \] \[ 90 \times 8 = 720 \] \[ 720 \times 7 = 5040 \] \[ 5040 \times 6 = 30240 \] \[ 30240 \times 5 = 151200 \] So, the number of different 6-digit combinations possible if no digit is repeated is 151200.