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 from 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 α × 5 6 , then α is equal to

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 from 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 α × 5 6 , then α is equal to,

MARKS App · Accepted Answer

Given total available characters is 10 and total digits are 5 , So, for 6 - digit password, the number of ways = 10 6 - 5 6 For 7 - digit password, the number of ways = 10 7 - 5 7 For 8 - digit password, the number of ways = 10 8 - 5 8 ∴ Total ways = 10 6 + 10 7 + 10 8 - 5 6 + 5 7 + 5 8 = 2 6 5 6 + 2 7 5 7 + 2 8 5 8 - 5 6 - 5 7 - 5 8 = 5 6 2 6 + 5 · 2 7 + 25 · 2 8 - 1 - 5 - 25 = 7073 · 5 6 Now on comparing with α × 5 6 we get α = 7073

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