How many two-character passwords are there that are made up of upper and lower c

Question

How many two-character passwords are there that are made up of upper and lower case letters and numbers, but where at least one of the characters is a number? There are 10 * 62 passwords where the first character is a number (prod), 62 * 10 passwords where the second character is a number (prod), and 10 * 10 passwords where both characters are numbers (prod). So there are 10 * 62 + 62 * 10 - 10 * 10 valid passwords in total (sub). // In set theory language, we call this the inclusion-exclusion principle: |A_1 union A_2| = |A_1| + |A_2| - |A_1 intersection A_2|.

Explanation / Answer

Dear Student Thank you for using Chegg !! Number of options for 2 character passwords = We divide the problem in two cases Case 1 :- When digit 1 is number and digit 2 can be both alphabet and number => 10 possibilities for digit 1 => For digit 2 nuber of possibilitites are 26 alphabets lower case 26 alphabets upper case 10 numbers Therefore total possibilities = 26 + 26 + 10 = 62 Total Number of password options under case 1 := 10*62 Case 1 :- When digit 1 is alphabet or number and digit 2 is number Similar to case 1 Total Number of password options under case 1 := 62*10 Now as can be observed the case when both the digits of password are numbers have been counted twice once under case 1 and secondly under case 2. Total Number of passwords when both charatcers are numbers = 10*10 Hence total number of passwords possible = 62*10 +10*62 - 10*10 1140 Solution

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