Final answer:
The number of possible five-digit entry codes for a door lock with ten buttons with repeated numbers allowed is 100,000. Without repetition, there are 30,240 unique entry codes possible.
Step-by-step explanation:
To answer the student’s question regarding the number of entry codes possible for a push-button door lock with ten buttons numbered 0 through 9:a. With repeated numbers allowedFor a five-digit code, with repeated numbers allowed, each digit in the sequence can be any of the ten possible numbers. This means we have 10 choices for the first digit, 10 choices for the second digit, and so on, resulting in 10^5 (ten to the power of five) possible combinations. That gives us a total of 100,000 entry codes.b. With no repeated digits allowed
For a five-digit code, without any repetition of numbers, the first digit can still be any of the 10 numbers. However, the second digit can now only be one of the remaining 9 numbers, the third digit one of 8 remaining numbers, and so on, until the last digit, which can be one of the 6 remaining numbers. This gives us a calculation of 10 x 9 x 8 x 7 x 6, which equals 30,240 possible unique entry codes.b) If no repeated numbers are allowed, the first digit has 10 options, the second digit has 9 options, the third digit has 8 options, the fourth digit has 7 options, and the fifth digit has 6 options. Therefore, there are a total of 10 x 9 x 8 x 7 x 6 = 30,240 possible entry codes without repeated digits.