5. In a computer game, Smješko has to get from the start to the finish, moving only east or south along the sides of a given grid. The grid consists of 15 equal squares, and the length of each side of the square is one step. (For example, from the start to the finish, he can move 3 steps east and then 5 steps south). In how many ways can Smješko reach the finish? Justify your answer.

![](https://cdn.mathpix.com/cropped/2024_06_03_f3cbcb3ab5c9e2a40cf2g-2.jpg?height=640&width=583&top_left_y=505&top_left_x=385)

## Ministry of Science and Education of the Republic of Croatia  Agency for Education and Upbringing  Croatian Mathematical Society

## COUNTY COMPETITION IN MATHEMATICS  March 1, 2023.
First solution.

We will briefly denote a move of one step to the east as I, and a move of one step to the south as $\mathrm{J}$.

Let's mark all the vertices of the square with points. From each point, Smješko can move to the east or south, and the number next to the point will count the different ways of moving from that place to the goal.

We will assign the number 1 to the points from which there is only 1 possible path to the goal.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-07.jpg?height=505&width=426&top_left_y=1895&top_left_x=244)

Consider the square in the lower right corner. From the last unmarked point to the goal, we can get there in 2 ways (IJ, JI). Therefore, we assign the number 2 to this point.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-08.jpg?height=500&width=426&top_left_y=230&top_left_x=244)

$1 \text{ POINT}$

From the point above the point we assigned the number 2, we can get to the goal in 3 ways (IJJ, JIJ, JJI). Therefore, we assign the number 3 to it.

The same applies to the point to the left of the point we assigned the number 2. From it, there are also 3 ways to get to the goal (IIJ, IJI, JII).

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-08.jpg?height=494&width=439&top_left_y=1012&top_left_x=243)

We conclude that the value assigned to each individual point is obtained by adding the values of the points that are one place to the east and one place to the south.

3 POINTS

By filling in the grid, we get the following image.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-08.jpg?height=479&width=357&top_left_y=1745&top_left_x=313)

Thus, the number of possible ways for Smješko to get from the start to the goal is 56.

Note: If a student correctly explains the conclusion and the correct solution without the image, they earn the right to 10 points. If a student has only the solution, they earn the right to 1 point. Partial conclusions with the correct solution can earn up to 8 points. The principle of "following the error" applies only with the correct conclusion (in the case of an incorrect sum of numbers on adjacent vertices of the square).

. $\text{TOTAL} 10$ POINTS

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