Question

The letters of the word RANDOM are written in all possible orders and these words are written out as in a dictionary then the rank of the word RANDOM is.
A.$614$
B.$615$
C.$612$
D.$616$

Answer 1

Here arrange the RANDOM word in alphabetical order, take one by one word as keeping the starting letter fixed and add all that words. After that, consider the word starting with R, Keeping the positions of RAD fixed we can rearrange the letters MNO . Similarly for RANDOM. Try it, you will get the answer.

Complete step-by-step answer:
Arranging RANDOM in alphabetical order gives us
ADMNOR.
Consider the words starting with A. If the starting letter is fixed as A, then the remaining $5$ letters can be rearranged in $5!$ ways.
Similarly with the words starting with letters D,M,N,O.
Hence $5\times 5!$
$=5\times 120=600$.
Now consider the words starting with R,
R−−−−−
In alphabetical order the first word starting with R will be
RADMNO
Keeping the positions of RAD fixed we can rearrange the letters MNO in $3!$ ways.
Similarly for RAMDNO.
Hence
$600+2(3!)=612$
The successive word will be
RANDMO and RANDOM.
Hence the rank of the word RANDOM is $612+2=614$.

Additional information:
Permutation relates to the act of arranging all the members of a set into some sequence or order. In other words, if the set is already ordered, then the rearranging of its elements is called the process of permuting. Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered.

Note: The concept behind permutation should be known. Also, permutation relates to the act of arranging all the members of a set into some sequence or order. A permutation is the choice of $r$ things from a set of $n$ things without replacement and where the order matters. The permutation has formula ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.