A large block, which has dimensions $n$ by 11 by 10 , is made up of a number of unit cubes and one 2 by 1 by 1 block. There are exactly 2362 positions in which the 2 by 1 by 1 block can be placed. What is the value of $n$ ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
Consider the block as $n$ layers each having 11 rows and 10 columns.

First, we consider positions of the 2 by 1 by 1 block which are entirely contained in one layer. In each layer, there are 9 possible positions for the 2 by 1 by 1 block in each row (crossing columns 1 and 2, 2 and 3, 3 and 4, and so on, up to 9 and 10), and there are 10 possible positions in each column (crossing rows 1 and 2, 2 and 3, 3 and 4, and so on, up to 10 and 11). Therefore, within each layer, there are $11(9)+10(10)=199$ positions for the 2 by 1 by 1 block. In the large block, there are thus $199 n$ positions of this type for the 2 by 1 by 1 block, since there are $n$ layers.

Next, we consider positions of the 2 by 1 by 1 block which cross between two layers. Since each layer has 110 blocks (in 11 rows and 10 columns) then there are 110 positions for the 2 by 1 by 1 block between each pair of touching layers. Since there are $n-1$ pairs of touching layers, then there are $110(n-1)$ positions of this type.

Thus, overall we have 2362 total positions, so

$$
\begin{aligned}
199 n+110(n-1) & =2362 \\
309 n-110 & =2362 \\
309 n & =2472 \\
n & =8
\end{aligned}
$$