Question

The letter of the word $RANDOM$ are written in all possible orders and these words are written out as dictionary then the rank of the word $RANDOM$ is

Answer 1

Hint: Use the following steps to reach the solution:
Step$1$: Write down the letters in alphabetical order.
Step $2$: Find the number of words that start with a superior letter.
Step$3$: Solve the same problem, without considering the first letter.
So by using this concept to reach the solution of the given problem.

Complete step-by-step answer:
It is given the word is $RANDOM$
In a dictionary the words are arranged in alphabetical order.
The words beginning with $A,D,M,N,O$ and $R$ are in the order.
Number of words starting with $A = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Number of words starting with $D = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Number of words starting with $M = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Number of words starting with $N = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Number of words starting with $O = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Number of words beginning with $R = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$,
But one of these words is $RANDOM$.
Firstly consider the words starting with $RAD$ and $RAM$.
Number of words beginning with $RAD = 3! = 3 \times 2 \times 1 = 6$
Number of words beginning with $RAM = 3! = 3 \times 2 \times 1 = 6$
These are $3!$ Words beginning with \[RAN\]one of these words is the word $RANDOM$ itself.
The first word beginning with \[RAN\] is the word \[RANDMO\]and the next word is $RANDOM$.
Hence we have to calculate that the rank of $RANDOM$,
Now we can add the possible way.
Therefore rank of $RANDOM$ =$5 \times 120 + 2 \times 6 + 2 = 614$

Rank of the word RANDOM is 614.

Note: 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 any meaning. Since there are $n!$ different words that are possible, a few simple tricks can minimize the efforts needed.