Question

The letters of the word WIFE are arranged in all possible ways and sorted in a dictionary manner. What is the position of the word WIFE in this arrangement?

Answer 1

Hint: In a dictionary the words are arranged in alphabetic order. We will first write the alphabetic order of the letters of the word WIFE. Then, find the number of possible arrangements of the word starting with E, F, and I. When there are $n$ distinct objects that need to be arranged in $n$ distinct ways, there are $n!$ ways possible. Similarly, write the number of words formed by W followed by E, F and I. Add all the number of possible ways to find the position of the word.

Complete step by step Answer:

As it is known that in the dictionary the words are arranged alphabetically.
The alphabetic order of the letters of the word WIFE is E, F, I, W.
First, we will find possible words starting with W.
If W is fixed, the other 3 words can be arranged in 3! Ways.
That, is there are \[3.2.1 = 6\] words starting with W.
Similarly, let us now fix F in the first place.
There are 6 words starting with F.
In a similar manner, there are 6 words starting with I.
Now, in the dictionary, when words starting with W are written, then initially the letter after W will be E.
If W and E are fixed there are 2 words and 2 possible ways.
Hence, the number of word starting with WE are $2! = 2.1 = 2$
Now, afterward, with WE, there will we words starting with WF
Again, there are 2 words that need to be arranged.
Hence, there are $2! = 2.1 = 2$ ways to arrange these words.
After all the words starting with WF, there will WIEF followed by WIFE.
We will now add all the words formed to find the position of the word WIFE.
$6 + 6 + 6 + 2 + 2 + 1 + 1 = 24$
So, we can say the position of the word WIFE is ${24^{th}}$

Note: We can also do this question by finding all the words possible from the word ‘WIFE’ and then subtracting the number of words formed after the required word, WIFE.
Here, there are 4 distinct letters that can be arranged in 4 possible ways, so, there are a total of $4! = 4.3.2.1 = 24$ and WIFE will be the last word when all the words will be arranged in a dictionary. Hence, the position of the word ‘WIFE’ will be ${24^{th}}$.