Question: If the letters of the word \[MASTER\] are permuted in all possible ways and the words thus formed ar...

Question

If the letters of the word $MASTER$ are permuted in all possible ways and the words thus formed are arranged in dictionary order, then find the rank of the word $MASTER$ .

Answer 1

Solution

As we need to find the all possible arrangements of the letters in the given word, we will use the concept of permutations and combinations.
We will find all the arrangements in dictionary order and will calculate the rank of the word MASTER.

Complete step-by-step answer:
Let us arrange the word $MASTER$ in alphabetic order.
$AEMRST$
Thus as per dictionary rank it is the first word.
Now let us keep “A” fixed, then the rest of the 5 words can be rearranged in 5! (=120) ways.
5! =120.
Thus the rank of the word starting with E will be greater than 120.
Now let us keep “E” as the first letter, the rest of the words can be rearranged in 5! (=120) ways.
Now, let us fix “M” as the first letter. Rearranging the next letters in alphabetical order gives
$MAERST$ .
Now let us keep “E” as the third letter, the next three letters “RST” can be permuted in 3! (=6) ways.
Let us fix the next letter “R” to occupy third position
$MAREST$ .
The next three letters “EST” can be permuted in 3! (=6) ways.
Let the fix the next letter “S” to occupy the third position, we get,
$MASERT$ .
Here we have got the first three letters fixed in the above word.
Let us fix “E” as the fourth letter, the remaining letters can be permuted in 2! That is 2 ways.
Let us fix “R” as the fourth letter, the remaining letters can be permuted in 2! That is 2 ways. We get
$MASRET$ ,
Let us fix “T” as the fourth letter, the remaining letters can be permuted in 2! That is 2 ways.
Here we get the required word $MASTER$ .
The rank is nothing but the sum of all possibilities.
The rank is $120 + 120 + 6 + 6 + 2 + 2 + 1 = 257$ .

Hence, The rank of the word MASTER is 257.

Note:
This tool calculates the rank of a given word when all the letters of that word are written in all possible orders and arranged in alphabetical or dictionary order. It supports words with repeating letters and non-repeating letters.