Rank Of A Word In A Dictionary - Practice Questions & MCQ

A common type of problem asked 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.

Rank of a word without repetition of letters

Example:

Find rank of MATHS.

Step 1: Write down the letters in alphabetical order.

The order will be $A, H, M, S, T$.

Step 2: Find the number of words that start with a superior letter.

Any word starting from A will be above MATHS. So, if we fix A at the first position, we have $4!=24$ words. (number of ways arranging $\mathrm{H}, \mathrm{M}, \mathrm{S}, \mathrm{T}$ ).

Similarly, there will be 24 words that will start with H .
Number of words start with MAH is $2!=2$

Number of words start with MAS is $2!=2$

Number of words start with MATHS is $1!=1$
Therefore, the overall rank of the word MATHS is $24+24+2+2+1=53$

Rank of a word with repetition of letters

Example:

Find the rank of a word INDIA in a dictionary made using its letters.

Write down the letters in alphabetical order, the order will be $\mathrm{A}, \mathrm{D}, \mathrm{I}, \mathrm{I}, \mathrm{N}$.
Number of words start with A is $4!/ 2!=12$ (We are dividing by $2!$ because I is repeating itself)
Number of words start with D is $4!/ 2!=12$
Number of words start with IA is $3!=6$ (number of ways arranging I, D, N)
Number of words start with ID is $3!=6$
Number of words start with II is $3!=6$
Number of words start with INA is $2!=2$
Number of words start with INDA is $1!=1$
Number of words start with INDIA is $1!=1$

Therefore, the overall rank of the word INDIA is $12+12+6+6+6+2+1+1=46$