Starting from her house to get to school, Júlia must walk 8 blocks to the right and 5 blocks up, as indicated in the figure below.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-01.jpg?height=361&width=528&top_left_y=1863&top_left_x=728)

She knows that there are many different ways to make the journey from home to school, always following the shortest path. As she is a very curious girl, she would like to always take different routes.

How many of these routes exist from Júlia's house to the school?

## List 2
No matter how Júlia walks from her house to school, she must travel 8 blocks to the right and 5 blocks up. A path connecting her house to the school is then a sequence of "block crossings," with 8 in the horizontal direction (to the right) and 5 in the vertical direction (up). Thus, to define a path, she only needs to decide the order in which she will make these crossings.

In this way, imagine 8 cards printed with the letter D and 5 cards printed with the letter C. Any permutation of these cards can be interpreted as a path to be traveled by Júlia. For example, the sequence of cards DDCDCCDDDDCDC defines the following path:

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-14.jpg?height=466&width=721&top_left_y=683&top_left_x=672)

To determine the number of ways to arrange these cards, we need to count how many different ways we can place 5 cards printed with the letter C in a row with 13 empty spaces, with the remaining 8 spaces in the row occupied by cards printed with the letter D.

Initially, we must choose one of the 13 empty spaces to place a letter C. Once this letter is placed, 12 empty spaces remain for the second letter C. Once this letter is placed, 11 empty spaces remain for the third letter, 10 spaces for the fourth letter, and finally 9 spaces for the fifth letter C. Now, once the five letters C are placed, any permutation of these letters among themselves does not change the distribution of the letters in the row. Since the number of permutations of five objects is \(5! = 120\), by the multiplication principle, the number of ways to arrange the 13 cards is

\[
\frac{13 \times 12 \times 11 \times 10 \times 9}{120} = 1287
\]

## List 2