GRE combination problem $877$.

Question

A computer password must be $5$ characters long. The first character must be a capital letter. The second character must be one of eight specified symbols. Each of the fourth and fifth characters can be any combination of capital or lowercase letters, digits, or symbols, where the symbols are from the $8$ specified symbols. How many different passwords can be made using these rules?

Why this is wrong? 26*10*8*70*69 first digit is selected from 26

second digit is select from 10 . third is selected from 8 and fourth is selected form 70 <26+26+8+10> , and fifth is selected from 69 , because all digits are different

Answer 1

I think your assumption that the last two characters need to be different from one another is unwarranted. Quoting from the problem statement:

"How many different passwords can be made using these rules?"

That doesn't dictate that the last two letters of a particular password cannot repeat, only that each password must be different from every other password.

The last two characters need not be different. Assuming # is one of the eight allowed symbols, the password, e.g. A9#mm is distinct from A9#mn and it is distinct from A9#nm. The fact that $m$ repeats gives a distinct password which fulfills the criteria for a password.

(Note also that the password A9### counts, because the first three characters follow criteria, and the last two are in the set $$\{\text{all uppercase letters in the alphabet}\} \cup \{\text{all lowercase letters in the alphabet}\}\cup \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \cup \{\text{eight allowed symbols one of which is #}\}$$

So you should have $26\cdot 10\cdot 8 \cdot 70 \cdot 70$.

Answer 2

$26\times10\times8\times70^2$. It is nowhere said that the digits must be different.