Final answer:
To find the number of distinct 3-character passwords with one numeral, one upper case letter, and one lower case letter, multiply the choices for each type of character. The calculation gives 10 (numerals) \(\times\) 26 (upper case letters) \(\times\) 26 (lower case letters) = 62 possible passwords. Option A is correct.
Step-by-step explanation:
The question asks us to calculate the number of distinct passwords possible when the password is composed of three characters: one numeral (0-9), one upper case letter, and one lower case letter from the English alphabet. There are 10 choices for the numeral (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), 26 choices for the upper case letter (A-Z), and 26 choices for the lower case letter (a-z). The total number of distinct passwords can be found by multiplying the number of choices for each character:
10 choices for the numeral
26 choices for the upper case letter
26 choices for the lower case letter
Therefore, the number of distinct passwords is 10 \(\times\) 26 \(\times\) 26, which equals 62 possible passwords. This result isn't listed as an option, so there seems to be an error in the provided choices. It's important to consider the order of characters as well.
If the order matters, then for each possible arrangement, there are three positions the numeral can take, the upper case letter can take either of the remaining two positions, and the lower case letter will take the last position, which means we should multiply our previous result by 3 factorial (3!). However, since we've already calculated the total number of combinations without considering order, there's no need to multiply by 3! again.