# seat arrangement possibilities

What could be possible arrangements for the below?

A, B, C, D and E are sitting on a bench. A is sitting next to B, C is sitting next to D, D is not sitting with E who is on the left end of the bench. C is on the second position from the right. A is to the right of B and E. A and C are sitting together.

• $E$ and $C$ have a fixed place. So there are only $3!=6$ arrangements to be tested... Commented Dec 4, 2014 at 11:49

According to the given information that

1. $E$ is on the left end of the bench and
2. $C$ is on the second position from the right you know that the $5$ places on the bench look like $$\underline{\color{blue}{E}}\,\underline{}\,\underline{}\,\underline{\color{blue}{C}}\,\underline{}$$

Now, consider the information that

1. $A$ is next to $B$ and that
2. $A$ is on the right of $B$ (and $E$)

which implies that the only possibility to place $A$ and $B$ is the following

$$\underline{E}\,\underline{\color{blue}{B}}\,\underline{\color{blue}{A}}\,\underline{C}\,\underline{}$$ This leaves one empty position to place $D$ on the right end $$\underline{E}\,\underline{B}\,\underline{A}\,\underline{C}\,\underline{\color{blue}{D}}$$

Note however that some of the information that you were given was not useful (or redundant) as for example that $A$ is on the right of $E$, that $E$ does not sit next to $C$ and that $A$ is next to $C$. These clues were implied by the rest.

$E$ is on the left end and so we first place $E$ and let the initial arrangement be $$E\star\star\star\star$$

After that we fix the position of $C$, then it becomes, $$E\star\star C\star$$ After that we fix the position of $D$, then we get,$$E\star\star CD$$

Then we fix $A$,

$$E\star A CD$$

Lastly $B$, $$\boxed{EB A CD}$$