A password contains exactly $6$ characters. Each character is either a lowercase letter $\{a,b,\dots,z\}$ or a digit $\{ 0,1,\dots,9\}$. A valid password should contain at least one digit. What is the total number of valid passwords?
- Here is an incorrect answer to the above question. Find the flaw in the argument. Let $P_i$ denote the number of passwords where the $i-th$ character is a digit, for $i\;\in \;1,\dots,6$.
$$P_1=10\cdot(36)^5$$
$$P_2=36\cdot 10\cdot(36)^4$$
$$\dots$$
$$P_6=(36)^5\cdot 10$$
Therefore, total number of valid passwords is $6\cdot 10\cdot (36)^5$ (the sum of the right hand sides above).
- What is the correct answer? Provide a justification for your answer. You do not need to simplify your expressions (for example, you can write $26^5, 5!, etc.$).