Let S be the set of all passwords which are six to eight characters long, where each chara…

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 repetition allowed. If the number of passwords in $S$ whose at least one character is a number from $\{1,2,3,4,5\}$ is $\alpha	imes5^6$, then $\alpha$ is equal to \_\_\_\_\_\_\_.

Answer: 7073

Accepted Answer

For length 6: total $10^6$, all-alphabet-only $5^6$, so at-least-one-number count $=10^6-5^6$.

For length 7: $10^7-5^7$. For length 8: $10^8-5^8$.

$$\text{Total} = (10^6+10^7+10^8) - (5^6+5^7+5^8)$$
$$= 5^6\left[(2^6+5\cdot2^7+25\cdot2^8) - (1+5+25)\right] = 5^6(64+640+6400-31) = 5^6\times7073$$

$$\alpha = 7073$$

Solutions · 1