The word “INEFFECTIVE” comprises 11 letters, with certain letters repeating. To determine the number of unique four-letter words that can be formed from these letters, we must consider the frequency of each letter and apply combinatorial principles accordingly.
Letter Frequency in “INEFFECTIVE”
Let’s first analyze the frequency of each letter in the word:
- E: 3 times
- F: 2 times
- I: 2 times
- N, C, T, V: 1 time each
This gives us a total of 7 distinct letters: E, F, I, N, C, T, V.
Categorizing Possible Four-Letter Word Combinations
To compute the total number of unique four-letter words, we consider different scenarios based on letter repetition:
1. All Letters Different (e.g., A, B, C, D)
- Selection: Choose any 4 distinct letters from the 7 available.
- Calculation:
2. Two Letters Same, Two Different (e.g., A, A, B, C)
- Selection:
- Choose one letter to repeat (E, F, or I):
- Choose two different letters from the remaining 6:
- Arrangement:
- Calculation:
3. Two Pairs of Identical Letters (e.g., A, A, B, B)
- Selection: Choose two letters to repeat (from E, F, I):
- Arrangement:
- Calculation:
4. Three Letters Same, One Different (e.g., A, A, A, B)
- Selection:
- Choose the letter to appear thrice (only E qualifies):
- Choose one different letter from the remaining 6:
- Arrangement:
- Calculation:
Total Number of Unique Four-Letter Words
Adding all the scenarios:
- All letters different: 840
- Two letters same, two different: 540
- Two pairs of identical letters: 18
- Three letters same, one different: 24
Total:
Conclusion
A total of 1,422 unique four-letter words can be formed from the letters of “INEFFECTIVE”, considering the repetitions and permutations of letters. This comprehensive approach ensures all valid combinations are accounted for.