Let S be the set of all passwords which are 6 | vedclass

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Question

(C) The total number of characters available is $5 	ext{ (alphabets)} + 5 	ext{ (numbers)} = 10$. The number of passwords of length $n$ is $10^{n}$.

Vedclass · Accepted Answer

$7073$

Vedclass · Answer

(C) The total number of characters available is $5 	ext{ (alphabets)} + 5 	ext{ (numbers)} = 10$. The number of passwords of length $n$ is $10^{n}$. The number of passwords of length $n$ containing no numbers (i.e.,only alphabets) is $5^{n}$. The number of passwords of length $n$ with at least one number is $10^{n} - 5^{n}$. For passwords of length $6, 7,$ and $8$,the total number of such passwords is: $(10^{6} - 5^{6}) + (10^{7} - 5^{7}) + (10^{8} - 5^{8})$ $= (10^{6} + 10^{7} + 10^{8}) - (5^{6} + 5^{7} + 5^{8})$ $= 10^{6}(1 + 10 + 100) - 5^{6}(1 + 5 + 25)$ $= 10^{6}(111) - 5^{6}(31)$ $= (2^{6} 	imes 5^{6}) 	imes 111 - 5^{6} 	imes 31$ $= 5^{6} 	imes (64 	imes 111 - 31)$ $= 5^{6} 	imes (7104 - 31)$ $= 5^{6} 	imes 7073$ Given this is equal to $\alpha 	imes 5^{6}$,we find $\alpha = 7073$.

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