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 $____________$

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

answer is 7073.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

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