Question

The rank of the word ‘FLOWER’ is
A) 165
B) 155
C) 145
D) None of the above

Answer 1

To solve this problem, we will use the fact that dictionary format uses the alphabetical format.
Thus, we will try to find the number of combinations of the words that can be made from the letters F, L, O, W, E and R which are in alphabetical order below the word, ‘FLOWER’. This will help us provide the rank of the word ‘FLOWER’.

Complete step by step solution:
Here the word given is ‘FLOWER’.
We are asked to find the rank of the word ‘FLOWER’ in the dictionary.
Now, in dictionary order, the letters of the word can be arranged as EFLORW. The first word in the dictionary will be EFLOWR.
Now, keeping the letter E fixed, the rest of the letters can be arranged in $5!$ ways.
After E comes F.
FELORW
So, keeping the letters F and E fixed, the rest of the letters can be arranged in $4!$ ways.
Then after comes F and L
FLEORW
Now, keeping F, L and E fixed, the rest of the letters can be arranged in $3!$ ways.
Then comes F, L and O
FLOEWR
So, keeping F, L, O and E fixed, the rest of the letters can be arranged in $2!$ ways.
Then comes F, L, O and R
FLOREW
So, keeping F, L, O and R fixed, the rest of the letters can be arranged in $2!$ ways.
Then comes the word we want to find
FLOWER
Keeping F, L, O, W and E fixed, the rest of the letters can be arranged in $1!$ way.
Thus, by the fundamental principle of adding, we get the rank of the word ‘FLOWER’ as
$5! + 4! + 3! + 2! + 2! + 1 \\\ = 120 + 24 + 6 + 2 + 2 + 1 \\\ =155$

Note:
Fundamental principle of addition:
The fundamental principle of addition states that, if we have X ways of doing something and Y ways of doing some other thing and those things cannot be done simultaneously, then the total number of ways to choose one of them is $X + Y$ .