5. In the figure below, you often see the capital letter A. A capital letter A can be written in many ways: for the two legs that come together at the top of the A, we can choose at what height to place the horizontal line between the legs. The legs of all letters must stand on the dotted line. Two of these capital As are bolded in the figure: on the left, an A with the line relatively low, and on the right, an A with the line very high.

How many capital As are there in this figure?
A) 55
B) 110
C) 220
D) 275
E) 550

![](https://cdn.mathpix.com/cropped/2024_04_17_f4c54c3a9ff27948c305g-2.jpg?height=474&width=532&top_left_y=631&top_left_x=1276)
A5. C) 220 We can make the letter A in 10 different heights. For each height, we keep track of how many there are. For height 1, there are 10 places where we can set the legs of the A; the horizontal line can only be in one place. For height 2, there are 9 places where the legs can be; the horizontal line can now be in 2 possible places. For height 3, there are 8 places where the legs can be; the horizontal line can be in 3 possible places. This pattern continues, and in total, we find $1 \cdot 10+2 \cdot 9+3 \cdot 8+4 \cdot 7+5 \cdot 6+6 \cdot 5+7 \cdot 4+8 \cdot 3+9 \cdot 2+10 \cdot 1=220$ possible letters A.