How many $5$-digit even palindromes are there?

A palindrome is a number whose digits read the same forwards and backwards, for example, $6336.$

In chess, a rook can attack any piece that lies on the same row or column as itself.

How many ways are there to place two identical rooks on a standard $8 \times 8$ checkerboard so that they are attacking each other?

How many ways are there for a checker to travel from the bottom left corner to the top right corner of this $5 \times 5$ board if it can only move right or up by one square each time, but it cannot pass through the center square?

At a school with 100 students, 40 of them attend English class, 50 of them attend Science class, and 55 of them attend Math class. In addition, 15 of them attend both English and Science, 17 of them attend both English and Math, and 20 of them attend both Math and Science. If every student in the school must attend at least one class, how many students attend all 3 classes?

How many ways are there for $3$ friends to share $10$ identical chocolates if everyone must get an whole number of chocolates, it is okay for someone to get zero chocolates, and all the chocolates must be handed out?

If $(x + 1)^9$ is fully expanded out, what is the sum of all of the coefficients?

Calculate:

$$\binom{3}{0} + \binom{4}{1} + \cdots + \binom{13}{10}$$

Here are some different ways to write $10$ as an ordered sum of positive integers less than or equal to $3.$

$$\begin{align*} 10 &= 2 + 2 + 2 + 2 + 2\\ 10 &= 2 + 3 + 3 + 2 \\ 10 &= 3 + 3 + 3 + 1\\ 10 &= 1 + 2 + 3 + 1 + 2 + 1\\ \end{align*}$$

How many different ways are there all together?

Six people are playing in a tennis tournament with 3 tennis courts. How many ways are there to pair them up into 3 pairs to play during the first session? It is not important which tennis courts they are playing on. Two sets of pairings are considered to be the same when the same people are playing each other.

How many ways are there to color each of the dots in this diagram using red, blue, and/or green such that there are no identical colors right next to each other? An example is shown below. It is not required that all three colors appear.