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# Number Theory

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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 "ARCH"?

A R C H

R R C H

C C C H

H H H H

Thanks!

Oct 18, 2023

#1
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There are 2 ways to start with the letter A: either from the first A in the array, or from the second A. To reach the second A, we must move down once. Once we have reached the first or second A, we can move in any direction to reach the letter R. There are 3 ways to do this: we can move down, right, or down and then right. Once we have reached the letter R, we can again move in any direction to reach the letter C. There are 3 ways to do this: we can move down, left, or left and then down. Once we have reached the letter C, the last step is to move up to reach the letter H, and there is only 1 way to do this. Therefore there are 2×3×3×1=18​ ways to spell the word "ARCH" in the array provided.

Oct 18, 2023
#3
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@bingboy, I had the same basic idea as you, but I think that you missed that the question for all distinct paths that spelled ARCH. Also, I don't get how there were two ways to start with A. Correct me if I'm wrong, though. Here's my solution:

Starting with first A in the array, we can make a tree diagram that explores all paths that go A to R to C to H.
From A, we can reach the 3 Rs in 3 ways by moving up, down, or diagonally.
From each of the 3 paths to the Rs, there are 2 paths from 2 Rs to a C and 5 paths from the other R to a C.
Now, even though we can arrive at the same C through different paths, we must take into account that the problem wants all distinct paths. From the 8 total C paths that we arrived at, we can finish our word with any of the 7 Hs. We end up with 2+3+2+3+5+3+2+2+3 distinct paths at the end, which equals an answer of 25.
It's much easier to visualize this by drawing a tree which I've hopefully attached. Oct 18, 2023