Finding the 2014 rearrangement of the word ORANGES?

Question

If someone is writing the 5040 arrangements of the letters in ORANGES in alphabetical order like: AEGNORS, AEGNOSR, AEGNROS, ... How would I find the 2014th rearrangement of this word?

Honestly this is a problem for my homework but I honestly don't know where to start with this problem. I know that there is a more logical way to go about this rather than writing out 2014 rearrangements of the word ORANGES.This community always helps me more than any TA or Professor ever will so this is why I am seeking help with this problem.

Hope you guys can help! Thanks

Answer 1

Lets work this out one letter at a time. Each letter gets a turn at being at the front of the word so $ \frac{1}{7} $ th of the words are going to start with each letter (or in another way $6!$). So first 720 words in the list are going to start with 'A'. The next 720 with 'E'.

$\frac{2014}{6!}$ gives us 2 with 574 as a remainder. So the first letter is 'G'.

You next take the remainder and continue the process with the remaining letters.

Answer 2

Working fairly slowly through the options:

$$\require{cancel} \begin{array}{|c|c|} \hline \text{Start seq} & \begin{array}{c} \text{Permutations of} \\ \text{remaining letters}\end{array} & \text{Takes us to line...} \\ \hline \text{A} & 6! = 720 & 720 \\ \hline \text{E} & 720 & 1440 \\ \hline \text{G} & 720 & \cancel{2160} \\ \hline \text{GA} & 5!=120 & 1560\\ \hline \text{GE} & 120 & 1680\\ \hline \text{GN} & 120 & 1800\\ \hline \text{GO} & 120 & 1920\\ \hline \text{GR} & 120 & \cancel{2040}\\ \hline \text{GRA} & 4!=24 & 1944\\ \hline \text{GRE} & 24 & 1968 \\ \hline \end{array}$$

etc...

I'd finish it, but I have a cold - have you seen my $2627$?

Answer 3

Write 2014 as follows: $$2014 = 2 \times 6! + 4 \times 5! + 3 \times 4! + 3 \times 3! + 2 \times 2!$$ Since the coefficient of $6!$ is 2, the first letter of the 2014th word is the third in the alphabetical order among $O, R, A, N, G, E, S$, and this is $G$. The coefficient of $5!$ is 4 and hence the next letter is the fifth among the remaining alphabets, and this is $R$. Proceeding similarly, we get the 2014th word as $GROSENA$