In the diagram, how many paths can be taken to spell "KARL"?
(A) 4
(B) 16
(C) 6
(D) 8
(E) 14

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-070.jpg?height=254&width=420&top_left_y=323&top_left_x=1297)
Solution 1

Starting at the "K" there are two possible paths that can be taken. At each "A", there are again two possible paths that can be taken. Similarly, at each "R" there are two possible paths that can be taken.

Therefore, the total number of paths is $2 \times 2 \times 2=8$.

(We can check this by actually tracing out the paths.)

## Solution 2

Each path from the $\mathrm{K}$ at the top to one of the L's at the bottom has to spell KARL.

There is 1 path that ends at the first $L$ from the left. This path passes through the first A and the first R.

There are 3 paths that end at the second $\mathrm{L}$. The first of these passes through the first A and the first R. The second of these passes through the first A and the second R. The third of these passes through the second $\mathrm{A}$ and the second $\mathrm{R}$.

There are 3 paths that end at the third L. The first of these passes through the first A and the second R. The second of these passes through the second A and the second R. The third of these passes through the second $\mathrm{A}$ and the third R.

There is 1 path that ends at the last L. This path passes through the last A and the last R.

So the total number of paths to get to the bottom row is $1+3+3+1=8$, which is the number of paths that can spell KARL.

ANswer: (D)