Let S be the set of all passwords which are six to eight

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Question:- Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is \alpha× 5^6, then α is equal to

Solution:- Number of password using {1, 2, 3, 4, 5} and {A, B, C, D, E} having 6 character = 10^6

Number of password using {1, 2, 3, 4, 5} and {A, B, C, D, E} having 7 character = 10^7

Number of password using {1, 2, 3, 4, 5} and {A, B, C, D, E} having 8 character = 10^8

Total  password = 10^6 + 10^7 + 10^8

Number of password using {1, 2, 3, 4, 5} having 6 character =  5^6

Number of password using {1, 2, 3, 4, 5} having 7 character =  5^7

Number of password using {1, 2, 3, 4, 5} having 8 character =  5^8

Number of password with no number = 5^6 + 5^7 + 5^8[/latex]

Total number of password with at least one number =(10^6 + 10^7 + 10^8)-(5^6 + 5^7 + 5^8)

= 10^6(1 + 10 + 100)-5^6(1 + 5 + 25)

= 10^6\times 111 - 5^6\times 31

= 5^6( 64\times 111 - 31)

= 5^6( 7104 - 31)

= 7073 \times 5^6

Exercise 7.3 ncert solutions maths class 11

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