2015 AMC 10A Problems/Problem 10

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Problem

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$.

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$ (Error compiling LaTeX. Unknown error_msg)


Solution

Observe that we can't begin a rearrangement with either $a$ or $d$, leaving $bcd$ and $abc$, respectively.

Starting with $b$, there is only one rearrangement: $bdac$. Similarly, there is only one rearrangement when we start with $c$: $cadb$.

Therefore, our answer must be $\boxed{\textbf{(C) }2}$.

See Also

2015 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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