CMI2020-B: 6

Question

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?

1. 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).

2. What is the correct answer? Provide a justification for your answer. You do not need to simplify your expressions (for example, you can write $26^5, 5!, etc.$).

Answer 1

Here is the solution, since they asked about password should contains at least 1 digit, which means it contains 1,2,3,4,5 or 6 digits

so we 1st find all passwords  which contains both alphabets as well as digits on that we remove passwords which contains only alphabets, then we will get required answer 

Answer 2