Question

The rank of the word “MADHUR” when arranged in dictionary order is:
A. 362
B. 360
C. 358
D. 356

Answer 1

Hint: Here, we have been given the word “MADHUR” and we have to arrange all the words formed by the letters in this word in the dictionary and tell the rank of the word “MADHUR” itself. For this, we will first arrange all the letters in this word in alphabetical order so that we could see how many words will come before the words starting from the letter ‘M’. We will calculate them by fixing the first position by the required letter and then finding the number of ways of arranging the remaining letters in the remaining positions. Then we will arrange the letters alphabetically after fixing the first position with the letter ‘M’ and then find the next combinations till we get the required word. Hence, we will get the position of the required word.

Complete step-by-step solution
Here, we have been given the word “MADHUR” and we have to arrange all the words formed by the letters in this word in the dictionary and tell the rank of the word “MADHUR” itself.
Now, the letters in this word as:
M
A
D
H
U
R
We know that alphabetically, all the words starting with ‘A’ will appear first.
Hence, we will first find the number of words starting from ‘A’.
Now, there are a total of 6 positions which have to be filled by 6 distinct letters among which the first position has been filled by ‘A’. This is shown as follows:
$\underset{\scriptscriptstyle-}{A}\ \_\ \_\ \_\ \_\ \_$
Now, we know that the number of ways of arranging ‘n’ distinct objects is given as n!.
We also know that the number of words starting with ‘A’ is equal to the number of ways the remaining letters can be arranged.
Thus, the number of words starting from ‘A’ are:
$\begin{align}
  & 5! \\
 & \Rightarrow 120 \\
\end{align}$
Now, after ‘A’ we know that alphabetically, the next set of words will all start from ‘D’.
Hence, we will calculate the number of words, which start with ‘D’.
Now, there are a total of 6 positions which have to be filled by 6 distinct letters among which the first position has been filled by ‘D’. This is shown as follows:
$\underset{\scriptscriptstyle-}{D}\ \_\ \_\ \_\ \_\ \_\ $
Hence, the number of words starting from the letter ‘D’ are:
$\begin{align}
  & 5! \\
 & \Rightarrow 120 \\
\end{align}$
Now, after ‘A’ and ‘D’, we know that alphabetically the next set of words will start from the letter ‘H’.
Hence, we will calculate the number of words, which start from ‘H’.
Now, there are a total of 6 positions which have to be filled by 6 distinct letters among which the first position has been filled by ‘H’. This is shown as follows:
$\underset{\scriptscriptstyle-}{H}\ \_\ \_\ \_\ \_\ \_\ $
Hence, the number of words starting from the letter ‘H’ are:
$\begin{align}
  & 5! \\
 & \Rightarrow 120 \\
\end{align}$
Now, we know that after ‘A’, ‘D’ and ‘H’, the next set of words will start from the letter ‘M’.
Now, if we arrange all the letters alphabetically after fixing the first position by ‘M’, we will get the word:
M A D H R U
Since we require the word “MADHUR”, we need to see what the next alphabetical arrangements will be.
Now, we will exchange the last two letters of the word now formed which will be the next word in the dictionary. It is given as:
M A D H U R
Now, this is the required word.
We know that its position is given as:
$5!+5!+5!+1+1$
This is given as:
$\begin{align}
  & 120+120+120+1+1 \\
 & \therefore 362 \\
\end{align}$
Hence, the word MADHUR is in the 362nd position.
Thus, option (A) is the correct option.

Note: These questions can also come in the opposite format. Here, we have been given a word and we have to find its position but we can also be given the position and find the word on that position. It is also done in a similar manner. Both the questions are equally important so practice them both.