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# Erm. halp

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How many ways can you spell "arch" in the figure below (I'm not sure if it'll show up properly lol)

A R C H

R R C H

C C C H

H H H H

I tried like literally counting it out, but I'm pretty sure there should be a better way than that lol

Jun 7, 2023

#1
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Once we have the first letter, we cannot use the initial A again. The second letter can be reached in 4 ways, as before, for a total of 4 ⋅ 4 = 16 ways. However, we cannot choose the same letter twice, so we must subtract the number of ways to spell the word "ARC" (there is only 1 way: right, right, down). Similarly, we cannot choose the same letter three times, so we must subtract the number of ways to spell the word "RHC" (there are 2 ways: right, down, right; down, right, right). Finally, we must exclude the word "RCH," which can be spelled only 1 way (right, right-down diagonal).

The total number of ways to spell "ARCH" is 16−1−2−1=12​.

Jun 7, 2023
#2
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1 - If you are allowed to repeat the digits, then you have

4 x 4 x 4 x 4 = 4^4 ==256 permutations.

2 - If you are not allowed to repeat the digits, then you have:

4 x 3 x 2 x 1 =4! =24 permutations.

Jun 7, 2023
#3
+44
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lmao. i think i forgot to mention that you can basically move right down left, horizontally and stuff, but not jumping around.

Jun 7, 2023
#6
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We are now permanently unable to use the letter A as the starting point. There are four possible routes to the second letter, just like the first, making the total number of routes 44 = 16. But since we can't pick the same letter twice, we have to take away the one possible method of spelling "ARC" (which is right, right, down). Also, since we can't pick the same letter three times, we need to take away the two possible ways of spelling "RHC" (right, down, right, and right, and right, down, right, and right). Last but not least, we can't include the letter combination "RCH," which has only one possible spelling (right, right-down diagonal).flying together

There are a total of 12 different ways to spell "ARCH" using the digits 16121.

Jun 14, 2023