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The Inclusion-Exclusion Principle is a handy tool in combinatorics that helps us accurately count the number of elements in a union of overlapping sets. When sets overlap, we might mistakenly count some elements more than once. This principle provides a systematic way to adjust for this overcounting.

Here's how it works: for two sets, you start by adding the sizes of the individual sets. However, because you've counted the overlapping elements twice, you must subtract the size of their intersection. For three sets like in this problem, it involves an extra layer. You add the sizes of the three sets, subtract the sizes of every pairwise intersection twice, then add back the size of the triple intersection that was subtracted too many times initially.

Considering the permutation problem where certain sequences of letters are not allowed, Inclusion-Exclusion allows us to first count each forbidden block, subtract instances where they overlap and adjust for any scenarios where three conditions overlap, ensuring we don't over-subtract and get a correct final count.

Combinatorics is the branch of mathematics dealing with combinations, arrangements, and selections of objects. It's fundamental for counting possibilities in structured ways, like determining the number of possible permutations or combinations given certain constraints.

A basic tool in combinatorics is the factorial operation, denoted as \(!\). A factorial \( n! \) is the product of all positive integers up to \( n \). For our problem with four distinct letters, the number of possible permutations (i.e., different orders) of those letters is \(4! = 24\). This is our starting point before considering any restrictions.

Combinatorics also often involves special counting techniques like using factorials to account for repeating elements or grouping elements into blocks to simplify calculations. Understanding how to manipulate these elements is key, especially when combined with other principles like Inclusion-Exclusion.

Permutation restrictions come into play when certain arrangements are not allowed within a set of ordered objects. In problems involving permutation restrictions, like the exercise presented, we are given specific rules about which sequences are not permissible.

For example, the constraint that "b does not follow a" means any sequence where b immediately follows a must be excluded from our count. Similarly, we need to exclude "c following b" and "d following c". This makes the puzzle more challenging as we must calculate permutations not just freely, but within the boundaries of these rules.

We approached these limitations by treating pairs or trios of letters (e.g., "ab," "bc," "cd," and "abc") as blocks. Each block acts as a single item in permutations, reducing the complexity. Totalizing these counts and correcting for overlaps with Inclusion-Exclusion, we can identify the number of permutations that meet all rules. By subtracting invalid arrangements from the total possible permutations, we find the valid permutation count that satisfies every restriction given.