Final answer:
For case a, without repetition of characters, there are 561600 possible five-character passwords. For case b, with repetition allowed, there are 1757600 possible passwords. This calculation considers 26 letters and 10 digits.
Step-by-step explanation:
To calculate the number of five-character passwords with the first three characters being letters and the last two being digits, we consider the English alphabet to have 26 letters and digits from 0 to 9 totaling 10 possible digits.
Case a: Repetition is not allowed
For the first character, there are 26 possibilities (A-Z). For the second character, as repetition is not allowed, there are 25 possibilities. For the third character, there are 24 possibilities left. For the fourth and fifth characters, as repetition is not allowed, there are 10 possibilities for the fourth character and only 9 left for the fifth character. The total number of non-repeating passwords is calculated by multiplying these possibilities together:
26 × 25 × 24 × 10 × 9 = 561600 possible passwords.
Case b: Repetition is allowed
When repetition is allowed, the situation is simpler. Each of the first three characters can be any of the 26 letters, and each of the last two can be any of the 10 digits, so we multiply these possibilities:
26 × 26 × 26 × 10 × 10 = 1757600 possible passwords.