Question 1210197: In the array below, in how many different ways can we start with the letter and move from letter to letter (horizontally, vertically, or diagonally), to spell the word "ARCS"?
A
RRR
CCCCC
SSSSSSS
Found 3 solutions by CPhill, greenestamps, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem.
**1. Visualize the Path**
We need to spell "ARCS" by moving from letter to letter in the array, horizontally, vertically, or diagonally.
```
A
RRR
CCCCC
SSSSSSS
```
**2. Tracing the Paths**
* **A:** There is only 1 starting "A."
* **R:** From "A," we can move to any of the three "R"s in the second row.
* **C:** From each "R," we can move to any of the five "C"s in the third row.
* **S:** From each "C," we can move to any of the seven "S"s in the fourth row.
**3. Counting the Ways**
* **A to R:** 1 (A) * 3 (R) = 3 ways.
* **R to C:** 3 (AR) * 5 (C) = 15 ways.
* **C to S:** 15 (ARC) * 7 (S) = 105 ways.
**Therefore, there are 105 different ways to spell "ARCS."**
Answer by greenestamps(13196)  (Show Source):
You can put this solution on YOUR website!
The AI solution from the other "tutor" uses an interpretation that is probably not what was intended.
I believe the array is supposed to look like this:
A
R R R
C C C C C
S S S S S S S
I further believe that the allowable diagonal moves from one row to the next are only to the letter either directly below or immediately to the left or to the right of the current letter.
In that case...
There is a single A in the first row;
From that A we can move to any of 3 R's in the second row;
From each R in the second row we can move to any of 3 C's in the third row;
From each C in the third row we can move to any of 3 S's in the last row.
Total number of ways to spell "ARCS": 3*3*3 = 27
ANSWER: 27
Analyzing the problem in a different way....
We need to move down from one row to the next at each move; we need to make 3 of those moves.
On each of those moves we have 3 choices: down and left, down, or down and right.
3 choices on each of 3 moves means a total of 3*3*3 = 27 total ways.
ANSWER: 27
Answer by ikleyn(52761)  (Show Source):
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