10. (40 points) Write "KOREAIMC" in eight lines as shown in Figure 5, where the first line has one K, the second line has two O's, and so on, with the last line having eight C's.
$$
\begin{array}{llllllll}
\mathbf{K} & & & & & & & \\
\mathbf{O} & \mathbf{O} & & & & & & \\
\mathbf{R} & \mathbf{R} & \mathbf{R} & & & & & \\
\mathrm{E} & \mathbf{E} & \mathrm{E} & \mathrm{E} & & & & \\
\mathrm{A} & \mathrm{A} & \mathbf{A} & \mathrm{A} & \mathrm{A} & & & \\
\mathrm{I} & \mathrm{I} & \mathrm{I} & \mathbf{I} & \mathrm{I} & \mathrm{I} & & \\
\mathrm{M} & \mathrm{M} & \mathbf{M} & \mathrm{M} & \mathrm{M} & \mathrm{M} & \mathrm{M} & \\
\mathrm{C} & \mathrm{C} & \mathrm{C} & \mathrm{C} & \mathrm{C} & \mathrm{C} & \mathrm{C} & \mathrm{C}
\end{array}
$$

Figure 5
Starting from the K at the top, move downward row by row to spell out "KOREAIMC". Each move can only be to the letter directly below or to a letter adjacent to the one directly below. The bold letters in Figure 5 represent one such path. If a letter is removed from Figure 5, such that the number of different remaining paths is only 516, identify the position of the removed letter.
10. First list the number table as shown in Figure 17, where each cell corresponds to a letter, and the number in the cell represents the number of different paths from the letter "K" to that letter. The cell at the very top is filled with 1. Starting from the second row, the number in each cell is the sum of the numbers in the cells from the row above that share a common vertex with it.
Then list

the number table as shown in Figure 18,
where each cell
also corresponds
to a letter, and the
number in the cell
indicates the number of different paths from that letter to the last row, filled in a similar manner to the table in Figure 17.

Thus, for each letter, the number table in Figure 17 gives the number of paths \( m \) from the letter "K" to that letter, while the number table in Figure 18 gives the number of paths \( n \) from that letter to the last row. If the letter is removed, the number of paths will be reduced by \( m \times n \). If the letter is not removed, the total number of different paths is 750. Since there are now only 516 paths, the total reduction is
$$
750 - 516 = 234 = 2 \times 3^{2} \times 13
$$

paths.
It is easy to see that only the numbers 13 and 26 in the fifth row of the number table in Figure 18 are multiples of 13.
At this point, \( 13 \times 9 = 117, 26 \times 9 = 234 \).
Therefore, the letter removed is the third letter "A" in the fifth row of Figure 5.