The problem is as follows:
The figure from below shows the word $\textrm{ROTATOR}$ arranged in a
peculiar way. How many ways can this word be read assuming the equal
least distance from one letter to another?.
The alternatives given in my book are as follows:
$\begin{array}{ll}
1.&\textrm{490 ways}\\
2.&\textrm{480 ways}\\
3.&\textrm{245 ways}\\
4.&\textrm{400 ways}\\
\end{array}$
I noticed that the word is palindromic, hence it can be read back and forth, thus this means that I should account for these possibilites.
In order to keep the right track for this purpose I used an auxiliary numbers atop the letters to account for these as shown in the diagram from below.
After doing all of that I reached the conclusion that:
$\textrm{ways}=(74+74+96)\times 2 =488$
But this doesn't appear in any of the alternatives. Did I mess up something or what? Can someone help me here? Please I require a step by step explanation as I feel lost if my method did worked out properly?
Best Answer
Here is a slightly easier argument.
Let's just count how many ways there are to go from an R to an A. The R's on one side give this:
$$R^1 \quad R^1 \quad R^1\\ O^1 \quad O^2 \quad O^2 \quad O^1 \\ T^3 \quad T^4 \quad T^3 \\ A^3 \quad A^7 \quad A^7 \quad A^3$$
Starting at the R's on the other end gives this:
$$A^4 \quad A^7 \quad A^7 \quad A^4 \\ T^1 \quad T^3 \quad T^4 \quad T^3 \quad T^1 \\ O^1 \quad O^2 \quad O^2 \quad O^1 \\ R^1 \quad R^1 \quad R^1$$
Adding those together, there are $7$, $14$, $14$, and $7$ ways to go from any R to each of the central A's. Conversely there are the same number of ways to go from each of those A's back to any R. Combining any R-to-A path with any A-to-R path from the same A we get $7\cdot7+14\cdot14+14\cdot14+7\cdot7 = 490$.