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 $\alpha \times 5^6$, then $\alpha$ is equal to ___

Answer 1

Answer: 7073

Required no. $=$ Total $-$ no character from $\{1,2,3$, $4,5\}$
$=\left(10^6-5^6\right)+\left(10^7-5^7\right)+\left(10^8-5^8\right)$
$=10^6(1+10+100)-5^6(1+5+25)$
$=10^6\times 111-5^6\times 31$
$=2^6\times 5^6\times 111-5^6\times 31$
$=5^6\left(2^6\times 111-31\right)$
$=5^6\times \underset{a}{\underbrace{7073}}$
$\therefore \alpha =7073$