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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"?

 

 Mar 21, 2024
 #1
avatar+394 
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The word "ARCH" has to contain the letter A.

There is only one letter A, therefore it much start there.

We can go hoizontally, diagonally, or vertically all once.

Therefore there are 3 ways to do this.

 Mar 21, 2024
 #2
avatar+1616 
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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.

 Mar 22, 2024

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