2015 AMC 10A Problems/Problem 10

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 $\boxed{\textbf{(C) }2}$ .

2015 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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All AMC 10 Problems and Solutions

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2015 AMC 10A Problems/Problem 10

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