Question

Make sure that you state your answer in a way that does not lose precision. Show all digits.

a. The Personal Identification Number (PIN) for your ATM card is four places long. Any number is permitted in each place. How many possible values of a PIN are there?

b. Your bank has decided to allow English alphabetic characters (upper or lower case are permitted) as well as any numeric digit in your PIN. How many possible PINs are there when that change is made?

c. In addition to b. above, your bank has decided to require that the first place of the PIN be an English upper case alphabetic character. How many possible PINs are there when that additional change is made?

d. In addition to b. and c. above, your bank has changed the PIN size to five places. Your PIN can now have numeric or English alphabetic characters (upper or lower case are allowed) in each place. How many possible PINs are there when that change is made?

Answer 1

a.

Since the Personal Identification Number (PIN) for the ATM card is four places long so each of these 4 places can be filled by any one of 0,1,2,3,4,...,9 in 10x10x10x10=10⁴=10000 ways.

b.

Since bank has decided to allow English alphabetic characters (upper or lower case are permitted) as well as any numeric digit in the PIN so each of these 4 places can be filled by any one of 10 digits, 26 alphabets (upper), 26 alphabets (lower) in

(10+26+26)⁴=14776336.

c.

Since bank has decided to require that the first place of the PIN be an English upper case alphabetic character so the first place can be filled by any one of 26 upper case alphabetic characters and each of remaining 3 places can be filled by any one of (10+26+26) characters. Hence total no. possible cases=26x(10+26+26)³= 6196528

d. When pin size is 5 places then total no. of cases=26x(10+26+26)⁴=384184736