Hello, Codeforces! The reason why I am writing this blog is that my ACM/ICPC teammate calabash_boy is learning this technique recently(he is a master in string algorithms,btw), and he wanted me to provide some useful resources on this topic. I found that although many claim that they do know this topic well, problems concerning inclusion-exclusion principle are sometimes quite tricky and not that easy to deal with. Also, after some few investigations, the so-called "Inclusion-Exclusion principle" some people claim that they know wasn't the generalized one, and has little use when solving problems. So, what I am going to pose here, is somewhat the "Generalized Inclusion-Exclusion Principle". Most of the describing text are from the graduate text book Graduate Text in Mathematics 238, A Course in Enumeration, and the problems are those that I encountered in real problem set, so if possible, I'll add a link to the real problem so that you can solve it by yourself. I'll start with the basic formula, one can choose to skip some of the text depending on your grasp with the topic.

Consider a finite set X and three subsets A, B, C, To obtain , we take the sum + + . Unless A, B, C are pairwise distinct, we have an overcount, since the elements of has been counted twice. So we subtract . Now the count is correct except for the elements in which have been added three times, but also subtracted three times. The answer is therefore

, or equivalently,

The following formula addresses the case applied to more sets.

The Restricted Inclusion-Exclusion Principle. Let A₁, A₂, ..., A_n be subsets of X. Then

This is a formula which looks familiar to many people, I'll call it The Restricted Inclusion-Exclusion Principle, it can convert the problem of calculating the size of the union of some sets into calculating the size of the intersection of some sets. It's not hard to prove the correctness of this formula, we can just check how often an element is counted in both sides. If , then it's counted once on either side. Suppose , and more precisely, that x is in exactly m of the sets A_i. The count on the left-hand side is 0, and on the right hand side, we have

for m ≥ 1, thus the equality holds.

Example 1. Let's see an example problem Co-prime where this principle could be applied: Given N, L, R, you need to compute the number of integers x in the interval [L, R] such that x is coprime with N, that is, gcd(x, N) = 1. There are 1 ≤ T ≤ 100 testcases. Constraints: 1 ≤ N ≤ 10⁹, 1 ≤ L ≤ R ≤ 10¹⁵.

The standard interpretation leads to the principle of inclusion-exclusion. Suppose we are given a set X, called the universe, and a set E = {e₁, e₂, ..., e_n} of properties that the elements of X may or may not process. Here we can define the properties as we like, such as e = " ≤ 5", , or even . Let A_i be the subset of elements that enjoy property e_i(and possibly others). Then is the number of elements that process none of the properties. Clearly, A_i1, A_i2, ..., A_it is the set of elements that possess the properties e_i1, e_i2, ..., e_it(and maybe others). Using the notation

we arrive at the principle of inclusion-exclusion.

Principle of Inclusion-Exclusion. Let X be a set, and E = {e₁, e₂, ..., e_n} a set of properties. Then

The formula becomes even simpler when N_{supseteq T} depends only on the size . We can then write N_{supseteq T} = N_≥ k for , and call E a homogeneous set of properties, and in this case N_= T = N_= k also depends only on the cardinality of T. Hence for homogeneous properties, we have

This is the very essence of Principle of Inclusion-Exclusion. Please make sure you understand every notation before you proceed. One can figure out, by letting , we arrive at the restricted inclusion-exclusion principle.

Example 2. This problem Character Encoding requires you to compute the number of solutions to the equation x₁ + x₂ + ... + x_n = k, satisfying that 0 ≤ x_i < m? Constraints: . Hint: the number of non-negative integer solutions of x₁ + x₂ + dots + x_n = k is given by .