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 $\alpha \times 5^{6}$, then $\alpha$ is equal to ___________.
Answer (integer)
7073
Solution
<p>If password is 6 character long, then</p>
<p>Total number of ways having atleast one number $= {10^6} - {5^6}$</p>
<p>Similarly, if 7 character long $= {10^7} - {5^7}$</p>
<p>and if 8-character long $= {10^8} - {5^8}$</p>
<p>Number of password $= ({10^6} + {10^7} + {10^8}) - ({5^6} + {5^7} + {5^8})$</p>
<p>$= {5^6}({2^6} + {5.2^7} + {25.2^8} - 1 - 5 - 25)$</p>
<p>$= {5^6}(64 + 640 + 6400 - 31)$</p>
<p>$= 7073 \times {5^6}$</p>
<p>$\therefore$ $\alpha = 7073$.</p>
About this question
Subject: Mathematics · Chapter: Permutations and Combinations · Topic: Fundamental Counting Principle
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