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 $α \times 5^{6}$ ,then $α$ is equal to $____________$

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answer is 7073.

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Detailed Solution

$R e q u i r e d p a s s w o r d s w i t h 6 c h a r a c t e r s = 10^{6} - 5^{6}, R e q u i r e d p a s s w o r d s w i t h 7 c h a r a c t e r s = 10^{7} - 5^{7}, R e q u i r e d p a s s w o r d s w i t h 8 c h a r a c t e r s = 10^{8} - 5^{8} R e q u i r e d n o . = T o t a l - n o c h a r a c t e r f r o m {1, 2, 3, 4, 5} = (10^{6} - 5^{6}) + (10^{7} - 5^{7}) + (10^{8} - 5^{8}) = 10^{6} (1 + 10 + 100) - 5^{6} (1 + 5 + 25)'' = 10^{6} \times 111 - 5^{6} \times 31 = 2^{6} \times 5^{6} \times 111 - 5^{6} \times 31 = 5^{6} (2^{6} \times 111 - 31) = 5^{6} \times 7073 ∴ α = 7073$

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