3. Given a rectangular grid of size 1 x 60. In how many ways can it be cut into grid rectangles of size 1 x 3 and 1 x 4?
3. Answer: 45665.

Solution. Let $x$ be the number of rectangles of size 1 x $3, y$ be the number of rectangles of size $1 \times 4$. Then $3 x+4 y=60$, where $x, y$ are non-negative integers. The value $\mathrm{x}=\frac{60-4 \mathrm{y}}{3}=20-\frac{4 \mathrm{y}}{3}$ is an integer if $\mathrm{y}=3 \mathrm{t}(\mathrm{t} \in \mathrm{Z})$, hence $\mathrm{x}=20 \cdot 4 \mathrm{t} \cdot(0 \leq t \leq 5)$. In the end, we get 6 solutions to the equation $3 x+4 y=60$:

$(20 ; 0),(16 ; 3),(12 ; 6),(8 ; 9),(4 ; 12),(0 ; 15)$. The number of ways to cut the given rectangle into $x$ rectangles of size 1 x 3 and $y$ rectangles of size 1 x 4 is $\mathrm{C}^{0}_{20}+\mathrm{C}^{3}_{19}+\mathrm{C}^{6}_{18}+\mathrm{C}^{9}_{17}+\mathrm{C}^{12}_{16}+\mathrm{C}^{15}_{15}=1+969+18564+24310+1820+1=45665$ ways.