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What is the total number of different -letter arrangements that can be formed using the letters in the word VEHEMENT?Answer:
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Q. What is the total number of different -letter arrangements that can be formed using the letters in the word VEHEMENT?Answer:
- Count Letters: Determine the number of times each letter appears in the word VEHEMENT. appears once, appears three times, appears once, appears once, appears once, and appears once.
- Total Arrangements: Calculate the total number of arrangements without considering the repetition of letters.Since the word VEHEMENT has letters, if all letters were unique, the number of different arrangements would be factorial ().
- Consider Repetition of E: Calculate the number of arrangements considering the repetition of the letter E. Since the letter E is repeated times, we need to divide the total number of arrangements by the factorial of the number of times E is repeated to avoid overcounting. So we divide by (which is the factorial of ).
- Calculate Permutations: Perform the actual calculation using the formula for permutations of a multiset.The number of different arrangements (permutations) is given by:Total arrangements = Now, calculate and :
- Final Answer: Divide the total number of unique arrangements by the number of repeated arrangements to get the final answer.Total arrangements =
