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

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 \underbrace{7073}_a $
$ \therefore \alpha=7073$