Rank of a Word in Dictionary

A common type of problem in many examinations is to find the 'rank' of a given word in a dictionary. What this means is that you are supposed to find the position of that word when all permutations of the word are written in alphabetical order. Of course, the words do not need to have any meaning. Since there are $n!$ different words that are possible , a few simple tricks can be used to minimise the effort needed.

Here we go. Let's start with an easy word.

Suppose that you are given a word in which none of the letters is repeated, and are asked to find the rank of the word in a dictionary. For example, if the word given is CAT, it will be very easy to find its rank. You first write down all possible combinations of the letters, which are CAT, CTA, ATC, TCA, ACT, TAC. Now, you arrange them in alphabetical order, which gives ACT, ATC, CAT, CTA, TAC, TCA.

CAT is the $3^\text{rd}$ in the above list, so the rank of the word CAT is $\boxed{3}$.

Now that we have learned the basic concept, let's try some harder examples.

Let's say that the word is $SBIPO$. With 5 letters, the number of all possible arrangements is $5!=120.$ So, it is not practical to write all of them down and find the rank of $SBIPO$. To solve problems like this, here is the process we need to follow:

• Step 1: Write down the letters in alphabetical order.
  The correct order will be $B, I, O, P, S$.

• Step 2: Find the number of words that start with a superior letter.
  Any word starting from $B$ will be above $SBIPO.$ So, if we fix $B$ at the first position, we have $4! = 24$ words.
  Similarly, there will be 24 words that will start with $I,$ 24 words that will start with $O,$ and 24 words that will start with $P.$
  So, the total number of words that do not start with $S$ and are above $SBIPO$ is $4\times 24 = 96.$

• Step 3: Solve the same problem, without considering the first letter.
  We need to find the rank of $BIPO$.
  The alphabetical order is $B$, $I$, $O$, $P$.
  $BIPO$ will be the word right after $BIOP$.

Therefore, the overall rank of the word $SBIPO$ is $96 + 2 = 98.\ _\square$

So, this is what "rank of a word in dictionary" means. Now, what follows is a bit tough.

Let's consider the word IBPSPO. As you can see, the alphabet P occurs twice in it. The process basically remains the same as above. However, there will be a slight difference in the way we calculate the answer.

• Step 1: Write down the letters in alphabetical order.
  The correct order is B, I, O, P, P, S.

• Step 2: Find the number of words that start with a superior letter.
  Number of words that start with B will be $\frac{5!}{2!} = 60.$ (We are dividing by 2 because P is repeating itself.)

• Step 3: Solve the same problem, without considering the first letter.
  We have to find the rank of BPSPO.
  This will be the same as the rank of PSPO.
  The words above PSPO are the three words starting with O (and ending with PPS, PSP, SPP).
  Also, PPOS, PPSO, and PSOP will be above PSPO.
  Thus, PSPO will be the $7^\text{th}$ word in the list.
  So, BPSPO will be the $7^\text{th}$ word in the list.

Therefore, the overall rank of the word IBPSPO is $60 + 7 = 67.\ _\square$