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Find the rank of the word MOTHER if all ...

Find the rank of the word MOTHER if all words with letters of MOTHER are written in alphabetical order

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To find the rank of the word "MOTHER" when all the permutations of its letters are arranged in alphabetical order, we can follow these steps: ### Step 1: Arrange the letters in alphabetical order The letters in "MOTHER" are: E, H, M, O, R, T. In alphabetical order, they are: - E, H, M, O, R, T ### Step 2: Count the words starting with letters before 'M' We need to count how many words can be formed starting with each letter that comes before 'M' in the alphabetical order. 1. **Words starting with 'E':** - Remaining letters: H, M, O, R, T (5 letters) - Number of arrangements = 5! = 120 2. **Words starting with 'H':** - Remaining letters: E, M, O, R, T (5 letters) - Number of arrangements = 5! = 120 ### Step 3: Count the words starting with 'M' Now we fix 'M' as the first letter and move to the second letter. 1. **Words starting with 'ME':** - Remaining letters: H, O, R, T (4 letters) - Number of arrangements = 4! = 24 2. **Words starting with 'MH':** - Remaining letters: E, O, R, T (4 letters) - Number of arrangements = 4! = 24 ### Step 4: Fix 'MO' as the first two letters Now we fix 'MO' and move to the third letter. 1. **Words starting with 'MOE':** - Remaining letters: H, R, T (3 letters) - Number of arrangements = 3! = 6 2. **Words starting with 'MOH':** - Remaining letters: E, R, T (3 letters) - Number of arrangements = 3! = 6 ### Step 5: Fix 'MOT' as the first three letters Now we fix 'MOT' and move to the fourth letter. 1. **Words starting with 'MOTE':** - Remaining letters: H, R (2 letters) - Number of arrangements = 2! = 2 2. **Words starting with 'MOTH':** - Remaining letters: E, R (2 letters) - Number of arrangements = 2! = 2 ### Step 6: Fix 'MOTHER' Finally, we have reached the word "MOTHER". ### Step 7: Calculate the total rank Now we sum up all the arrangements counted: - Starting with 'E': 120 - Starting with 'H': 120 - Starting with 'ME': 24 - Starting with 'MH': 24 - Starting with 'MOE': 6 - Starting with 'MOH': 6 - Starting with 'MOTE': 2 - Starting with 'MOTH': 2 Total = 120 + 120 + 24 + 24 + 6 + 6 + 2 + 2 = 304 Since "MOTHER" is the next word, we add 1 to the total to find its rank: - Rank of "MOTHER" = 304 + 1 = **305** ### Final Answer The rank of the word "MOTHER" is **305**.
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