![]() ![]() Let's say you are tasked with creating a 10 10 10 character-long password with uppercase and lowercase letters. This will be much easier to understand with a couple of examples. As you can imagine, for a higher number of conditions (e.g., at least one number and one uppercase letter), the groups of incorrect passwords increase in number. To do so, we use the inclusion-exclusion principle, a tool that stems from set theory. We take all the possible permutations, with repetitions that contain the desired type of character, and then we subtract the permutations that don't respect the condition. How do we write this condition in numbers? ![]() We know it's frustrating when this happens, but trust us, compared to a simple password made of only lowercase letters, this improves the safety of your accounts many times. Before this, we need to calculate the password permutations when a condition of the type at least one uppercase character/number/symbol sets in. In the next section, you will see how quickly the number of permutations can grow. It's not hard to imagine that the result of the example above comes from the division of two factorials: If this multiplication of consecutive numbers rung a bell, it's because the formula for the number of permutations uses the factorial: n ! = 1 ⋅ 2 ⋅ 3 ⋅. ![]() And after Step 3, you'd have a distinct three-item set from among 10 ⋅ 9 ⋅ 8 = 720 10 \cdot 9 \cdot 8 = 720 10 ⋅ 9 ⋅ 8 = 720 possible outcomes. After Step 2, you'd have one pair from among 9 ⋅ 10 = 90 9 \cdot 10 = 90 9 ⋅ 10 = 90 possible pairs.
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