3. A set of 55 dominoes contains all possible combinations of two numbers from 0 to 9, including dominoes with the same number on both ends. (In the image, three dominoes are shown: a domino with the numbers 3 and 4, a domino with the numbers 0 and 9, and a domino with the number 3 on both ends.) How many dots are there in total in the entire set of dominoes?

![](https://cdn.mathpix.com/cropped/2024_06_04_841f52b4ffb037ad799bg-15.jpg?height=326&width=782&top_left_y=168&top_left_x=455)
3. Let's first calculate the total number of dots on the dominoes where two different numbers appear. Consider all ordered pairs of different numbers from 0 to 9. Each number appears exactly 9 times in the first coordinate (for all possible values of the second coordinate different from that number), so the sum of all first coordinates is $9 \cdot (0+1+2+\cdots+9) = 9 \cdot \frac{9 \cdot 10}{2} = 405$; similarly, the sum of all second coordinates is also 405, so the sum of all numbers that appear in these ordered pairs is 810. Note that the dominoes where two different numbers appear can be represented by such ordered pairs, where each domino has been counted twice (the domino with numbers $a$ and $b$ has been counted both as the pair $(a, b)$ and as the pair $(b, a)$); thus, the total number of dots on these dominoes is exactly half of the value calculated above, i.e., it is 405.

The total number of dots on the dominoes where the same number appears on both sides is $2 \cdot (0+1+2+\cdots+9) = 2 \cdot \frac{9 \cdot 10}{2} = 90$. Therefore, the solution to the problem is $405 + 90 = 495$.