Consider a $21 \times 17$ rectangular region. This region is to be tiled using tiles of the two types shown in ./mingyin/square1.png (The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping. What is the minimum number of tiles required to tile the region?
Relevant Theorem: Counting, in the context of combinatorics, is a branch of mathematics that deals with the enumeration, arrangement, and selection of objects or elements in a set. It involves finding the number of ways to perform a specific task, such as arranging items in a certain order, selecting a subset of items from a larger set, or distributing items among different groups.

Combinatorics uses various techniques and principles to solve counting problems, including:

1. The Rule of Sum: If there are m ways to perform one task and n ways to perform another task, and these tasks cannot be performed simultaneously, then there are m + n ways to perform either task.

2. The Rule of Product: If there are m ways to perform one task and n ways to perform another task, and these tasks can be performed independently, then there are m * n ways to perform both tasks.

3. Permutations: A permutation is an arrangement of objects in a specific order. The number of permutations of n objects is given by n! (n factorial), which is the product of all positive integers up to n.

4. Combinations: A combination is a selection of objects without regard to their order. The number of combinations of n objects taken r at a time is given by the binomial coefficient, denoted as C(n, r) or "n choose r," and calculated as C(n, r) = n! / (r! * (n-r)!).

5. The Pigeonhole Principle: If n items are placed into m containers, and n > m, then at least one container must contain more than one item. This principle is used to prove the existence of certain arrangements or selections.

6. Inclusion-Exclusion Principle: This principle is used to count the number of elements in the union of multiple sets by considering the overlaps between the sets. It involves adding the sizes of individual sets and subtracting the sizes of their intersections.

Counting problems are common in various fields, including probability theory, statistics, computer science, and cryptography. Combinatorics helps in solving these problems by providing systematic methods and techniques for counting and organizing objects.
99