Question

How many different license passwords can one make if said password must contain exactly 6 characters, two of which are distinct numbers, another of which must be an uppercase letter, and the remaining 3 can be any digit or letter (upper- or lower-case) such that there are no repetitions of any characters in the password?

• A. 231
• B. 456426360
• C. 219
• D. 619652800
• E. 365580800

Hint:

When no repetition is allowed, always reduce the available character set for each subsequent pick.

Answer 1

The Correct Option is B

Step 1: Choices for 2 distinct digits.
There are 10 digits, so ways = \(\binom{10}{2} = 45\).
Step 2: Choices for 1 uppercase letter.
There are 26 uppercase letters.
Step 3: Remaining 3 characters from 62 total (26 uppercase + 26 lowercase + 10 digits) minus used.
Used = 3 characters. Remaining = 59. So ways = \(P(59,3) = 59 \times 58 \times 57\).
Step 4: Multiply possibilities.
Total = \(45 \times 26 \times (59 \times 58 \times 57)\). After simplifying, it equals 456426360.
Final Answer: \[ \boxed{456426360} \]