A computer uses passwords that consist of the 26 letters (a-z) and the 10 number

Question

A computer uses passwords that consist of the 26 letters (a-z) and the 10 numbers (0-9). How many dierent passwords are possible if:

(a) Exactly 6 characters (letters or numbers) are required, and the letters are not case-sensitive (i.e., no dierence between upper- and lower-case)

(b) Passwords must have 4 letters followed by 2 numbers, again without being case-sensitive

(c) Passwords have 4-6 characters, and they are case-sensitive

(d) Passwords have 6 characters, not case-sensitive, but no character can appear more than once

(e) A student wrote their password on a piece of paper, cut out each letter (all lowercase a-z), mixed them, and placed them in front of you. There are 6 pieces. How many possible passwords could the student have

Explanation / Answer

a)

There are 36 options for each character (26+10).

Hence, there are 36^6 = 2176782336 WAYS [ANSWER]

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b)

There are 26 options for the first 4 characters, and 10 options for the last 2. Hence, there are

26*26*26*26*10*10 = 45697600 [ANSWER]

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c)

So we sort of have 52 letters, as it is now case sensitive.

Hence, there are 52+10 = 62 ways to choose each character.

Hence, as it can be 4-6 characters long,

62^4+ 62^5 + 62^6 = 57731144752 ways [ANSWER]

***************

d)

There are 62P6 = 44261653680 ways [ANSWER]

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e)

There are 6! = 720 ways [ANSWER]

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