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Question: The letter of the word \(RANDOM\) are written in all possible orders and these words are written out...

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

Explanation

Solution

Use the following steps to reach the solution:
Step11: Write down the letters in alphabetical order.
Step 22: Find the number of words that start with a superior letter.
Step33: 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 RANDOMRANDOM
In a dictionary the words are arranged in alphabetical order.
The words beginning with A,D,M,N,OA,D,M,N,O and RR are in the order.
Number of words starting with A=5!=5×4×3×2×1=120A = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Number of words starting with D=5!=5×4×3×2×1=120D = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Number of words starting with M=5!=5×4×3×2×1=120M = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Number of words starting with N=5!=5×4×3×2×1=120N = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Number of words starting with O=5!=5×4×3×2×1=120O = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Number of words beginning with R=5!=5×4×3×2×1=120R = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120,
But one of these words is RANDOMRANDOM.
Firstly consider the words starting with RADRAD and RAMRAM.
Number of words beginning with RAD=3!=3×2×1=6RAD = 3! = 3 \times 2 \times 1 = 6
Number of words beginning with RAM=3!=3×2×1=6RAM = 3! = 3 \times 2 \times 1 = 6
These are 3!3! Words beginning with RANRANone of these words is the word RANDOMRANDOM itself.
The first word beginning with RANRAN is the word RANDMORANDMOand the next word is RANDOMRANDOM.
Hence we have to calculate that the rank of RANDOMRANDOM,
Now we can add the possible way.
Therefore rank of RANDOMRANDOM =5×120+2×6+2=6145 \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!n! different words that are possible, a few simple tricks can minimize the efforts needed.