11.3. A square with a side of 4 cm is divided by three parallel horizontal and three parallel vertical lines into 16 smaller squares with a side of 1 cm. The sides of these smaller squares, including those located on the boundary of the larger square, will be called unit segments. In how many ways can an orientation be assigned to each of the 40 unit segments so that the total sum of all 40 vectors is equal to 0? The answer can be given in the form of a formula, without necessarily reducing it to a number.
Answer. $C_{20}^{10} \cdot C_{20}^{10}$.

Solution. In total, there will be 20 horizontal and 20 vertical segments. If we consider them to be located along the coordinate axes OX and OY, respectively, and denote the number of positively oriented horizontal and vertical segments by $x$ and $y$ respectively, then the coordinates of the sum of all the obtained vectors will be $(2x-20, 2y-20)$. Therefore, for the sum of all vectors to be zero, it is necessary and sufficient that exactly half of the horizontal and exactly half of the vertical segments are oriented positively, and the rest negatively. Moreover, it does not matter which specific ones. There are $C_{20}^{10}$ ways to choose 10 out of 20 positively oriented horizontal segments and $C_{20}^{10}$ ways to choose 10 out of 20 positively oriented vertical segments, and these choices are independent of each other, so the answer will be the number $C_{20}^{10} \cdot C_{20}^{10}$. Grading criteria. Proof that exactly half of the horizontal and exactly half of the vertical segments are oriented positively, and the rest negatively: 3 points.