Question

If all the letters of the word 'RAPID' are arranged in all possible manner as they are in a dictionary
then find the rank of the word 'RAPID'.
15​

Answer 1

Answer:

First of all, arrange all letters of given word alphabetically : 'ADIPR'

Total number of words starting with A _ _ _ _ = 4! = 24

Total number of words starting with D _ _ _ _ = 4! = 24

Total number of words starting with I _ _ _ _ = 4! = 24

Total number of words starting with P _ _ _ _ = 4! = 24

Total number of words starting with RAD _ _ = 2! = 2

Total number of words starting with RAI _ _ = 2! = 2

Total number of words starting with RAPD _ = 1

Total number of words starting with RAPI _ = 1

\ Rank of the word RAPID = 24 + 24 + 24 + 24 + 2 + 2 + 1 + 1 = 102 Ans.

Answer 2

To Find:

The rank of the word 'RAPID', if all the letters of the word 'RAPID' are arranged in all possible manners as they are in a dictionary.

Solution:

The words in a dictionary are written in alphabetical order.

The letters of the word 'RAPID' arranged alphabetically will be 'ADIPR'.

1. Starting with 'A', the remaining letters D, I, P, R can be arranged in $\[4! = 24\]$ ways. So, the number of words starting with $\[{\rm{A}}\_\_\_\_ = 24\]$.

2. Similarly, the number of words starting with  $\[{\rm{D}}\_\_\_\_ = 24\]$.

3. Number of words starting with $\[{\rm{I}}\_\_\_\_ = 24\]$

4. Number of words starting with $\[{\rm{P}}\_\_\_\_ = 24\]$

5. Now, the number of words starting with $\[{\rm{R}}\_\_\_\_$ is also $24$. Out of these words, one word is 'RAPID'. Therefore,

Number of words starting with $\[{\rm{RAD\_\_}} = 2! = 2\]$

Number of words starting with $\[{\rm{RAI\_\_}} = 2! = 2\]$

6. Thus, so far $24+24+24+24+2+2=100$ words have been formed.

7. Now the words starting with $\[{\rm{RAP\_\_}}$ will appear. The first word beginning with 'RAP' is 'RAPDI' which is the $\[101{\rm{st}}\]$ word and the next word is 'RAPID' itself. Thus, 'RAPID' is the $\[102{\rm{nd}}\]$ word.

Hence, the rank of the word 'RAPID' is $102$.

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