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If the letters of the word CORONA are ar...

If the letters of the word CORONA are arranged in all possible ways and these words are written in order of a dictionary, then the word CORONA appears at the serial number

A

108

B

110

C

106

D

112

Text Solution

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The correct Answer is:
To find the serial number of the word "CORONA" when all the letters are arranged in alphabetical order, we can follow these steps: ### Step 1: Identify the letters and their frequency The letters in "CORONA" are C, O, R, O, N, A. The frequency of each letter is: - C: 1 - O: 2 - R: 1 - N: 1 - A: 1 ### Step 2: Arrange the letters in alphabetical order The alphabetical order of the letters is: A, C, N, O, O, R. ### Step 3: Count the arrangements before "CORONA" We need to count how many arrangements come before "CORONA" in alphabetical order. #### 3.1: Count arrangements starting with 'A' If 'A' is the first letter, the remaining letters are C, O, O, R, N. The number of arrangements is calculated as follows: \[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \] (We divide by \(2!\) because 'O' is repeated.) #### 3.2: Count arrangements starting with 'C' Next, we consider arrangements starting with 'C'. The remaining letters are A, O, O, R, N. ##### 3.2.1: Count arrangements starting with 'CA' If 'CA' is the first part, the remaining letters are O, O, R, N. The number of arrangements is: \[ \text{Arrangements} = \frac{4!}{2!} = \frac{24}{2} = 12 \] ##### 3.2.2: Count arrangements starting with 'CO' If 'CO' is the first part, the remaining letters are A, O, R, N. The number of arrangements is: \[ \text{Arrangements} = 4! = 24 \] ##### 3.2.3: Count arrangements starting with 'C' Now we consider arrangements starting with 'C', followed by 'N': - 'C', 'N' followed by A, O, O, R. ###### 3.2.3.1: Count arrangements starting with 'CNO' If 'CNO' is the first part, the remaining letters are A, O, R. The number of arrangements is: \[ \text{Arrangements} = 3! = 6 \] ###### 3.2.3.2: Count arrangements starting with 'C' Now we consider arrangements starting with 'COR': - 'COR' followed by A, O, N. ###### 3.2.3.2.1: Count arrangements starting with 'CORO' If 'CORO' is the first part, the remaining letters are A, N. The number of arrangements is: \[ \text{Arrangements} = 2! = 2 \] ### Step 4: Calculate the total arrangements before "CORONA" Now we sum all the arrangements: - Arrangements starting with 'A': 60 - Arrangements starting with 'CA': 12 - Arrangements starting with 'CO': 24 - Arrangements starting with 'CNO': 6 - Arrangements starting with 'CORO': 2 Total arrangements before "CORONA": \[ 60 + 12 + 24 + 6 + 2 = 104 \] ### Step 5: Find the rank of "CORONA" The rank of "CORONA" is the total arrangements before it plus 1 (for itself): \[ \text{Rank of CORONA} = 104 + 1 = 105 \] ### Final Answer The word "CORONA" appears at serial number **105**. ---
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