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The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is ________. (in integers)
Correct answer is '120'. Can you explain this answer?
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The number of words (with or without meaning) that can be formed from ...
Understanding the Problem
To find the number of arrangements of the letters in "LETTER" where vowels (E, E) do not come together, we can use complementary counting.
Total Arrangements of "LETTER"
- The word "LETTER" has 6 letters: L, E, T, T, E, R.
- The total arrangements are calculated using the formula for permutations of multiset:
- Total arrangements = 6! / (2! * 2!)
- This accounts for 2 E's and 2 T's being indistinguishable.
- Total arrangements = 720 / 4 = 180.
Arrangements with Vowels Together
- Treat the vowels (E, E) as a single unit, which we can denote as (EE).
- This gives us the new set of units: (EE), L, T, T, R (total of 5 units).
- The arrangements of these 5 units are:
- Arrangements = 5! / (2!) = 120.
- Here, we divide by 2! for the indistinguishable T's.
Calculating Vowels Not Together
- To find the arrangements where vowels do not come together, subtract the arrangements where vowels are together from the total arrangements:
- Arrangements where vowels are not together = Total arrangements - Arrangements with vowels together.
- Hence, Arrangements = 180 - 120 = 60.
Conclusion
The correct answer for the number of arrangements of the letters in "LETTER" where vowels never come together is 60.
To find the number of arrangements of the letters in "LETTER" where vowels (E, E) do not come together, we can use complementary counting.
Total Arrangements of "LETTER"
- The word "LETTER" has 6 letters: L, E, T, T, E, R.
- The total arrangements are calculated using the formula for permutations of multiset:
- Total arrangements = 6! / (2! * 2!)
- This accounts for 2 E's and 2 T's being indistinguishable.
- Total arrangements = 720 / 4 = 180.
Arrangements with Vowels Together
- Treat the vowels (E, E) as a single unit, which we can denote as (EE).
- This gives us the new set of units: (EE), L, T, T, R (total of 5 units).
- The arrangements of these 5 units are:
- Arrangements = 5! / (2!) = 120.
- Here, we divide by 2! for the indistinguishable T's.
Calculating Vowels Not Together
- To find the arrangements where vowels do not come together, subtract the arrangements where vowels are together from the total arrangements:
- Arrangements where vowels are not together = Total arrangements - Arrangements with vowels together.
- Hence, Arrangements = 180 - 120 = 60.
Conclusion
The correct answer for the number of arrangements of the letters in "LETTER" where vowels never come together is 60.
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Community Answer
The number of words (with or without meaning) that can be formed from ...
LETTER
Vowels = EE, Consonants = LTTR
_L_T_T_R_
Total number of words = 4! / 2! × 5C2 × 2! / 2! = 12 × 10 = 120
Vowels = EE, Consonants = LTTR
_L_T_T_R_
Total number of words = 4! / 2! × 5C2 × 2! / 2! = 12 × 10 = 120
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The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is ________. (in integers)Correct answer is '120'. Can you explain this answer?
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