How many times does the word BRAZIL appear in the figure on the side? Only count the word by joining letters that are written in adjacent squares.

|  |  |  |  | $\bar{B}$ | R |
| :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  | $\bar{B}$ | $\bar{R}$ | $\bar{A}$ |
|  |  | $B$ | $\bar{R}$ | $\bar{A}$ | $\bar{S}$ |
|  | $\bar{B}$ | $\mathrm{R}$ | $\mathrm{A}$ | $\bar{S}$ | $T$ |
| $B$ | $\mathrm{R}$ | $\mathrm{A}$ | $\mathrm{S}$ | $T$ | $\mathrm{~L}$ |
To read the word BRASIL, we must follow a path that starts at a letter B and ends at a letter L. Observe that the path to be followed is composed successively of horizontal moves to the right and vertical moves downward. Thus, let us represent these paths by sequences of letters $\mathrm{H}$ (meaning move to the right) and letters $\mathrm{V}$ (meaning move downward).

Let's look at two examples:

(i) We start at B in the second line (from top to bottom) and follow the path VHVVV.

(ii) We start at B in the third line and follow the path HVVHH.

To solve the problem, we need to count how many paths start with B and end with L. For this, we need to list these paths. Let $\mathcal{C}_{j}$ be the number of such paths starting in line $j$, where $j$ varies from 1 to 6:

Line 1: $V V V V V \leadsto \mathcal{C}_{1}=1$;

Line 2: HVVVV, VHVVV, VVHVV, VVVHV, VVVVH $\leadsto \mathcal{C}_{2}=5$;

Line 3: HHVVV, HVHVV, HVVHV, HVVVH, VHHVV, VHVHV, VHVVH, VVHHV, VVHVH, VVVHH $\leadsto \mathcal{C}_{3}=10$;

Line 4: HHHVV, HHVHV, HHVVH, HVHHV, HVHVH, HVVHH, VHHHV, VHHVH, VHVHH, VVHHH $\leadsto \mathcal{C}_{4}=10$;

Line 5: HHHHV, HHHVH, HHVHH, HVHHH, VHHHH $\leadsto \mathcal{C}_{5}=5$;

Line 6: $\mathrm{HHHHH} \leadsto \mathcal{C}_{6}=1$.

Therefore, the word BRASIL appears

$$
\mathcal{C}_{1}+\mathcal{C}_{2}+\mathcal{C}_{3}+\mathcal{C}_{4}+\mathcal{C}_{5}+\mathcal{C}_{6}=1+5+10+10+5+1=32
$$

times in the figure (Try to understand the symmetry: $\mathcal{C}_{1}=\mathcal{C}_{6} ; \mathcal{C}_{2}=\mathcal{C}_{5}$ and $\mathcal{C}_{3}=\mathcal{C}_{4}$).