Math Problem Statement
Solution
Let’s address the problem step-by-step for each part of the question:
The word: SELECTION
- It has 9 letters in total.
- Letter distribution:
- S, L, C, T, N → occur once each.
- E → occurs twice.
- I → occurs once.
- O → occurs once.
Now, we calculate arrangements for each case.
(i) No restriction
The total number of arrangements of the 9 letters, accounting for the repetition of the letter E:
(ii) It must begin and end with the letter "E"
If the arrangement must start and end with "E", these two positions are fixed. The remaining 7 letters (S, L, C, T, N, I, O) need to be arranged.
The number of arrangements is:
(iii) All the vowels must be separated
The vowels in "SELECTION" are E, E, I, and O (4 vowels). To ensure that no two vowels are adjacent, arrange the consonants (S, L, C, T, N) first, and then place the vowels in the gaps.
- Arrange the consonants: There are 5 consonants (S, L, C, T, N), so:
- Gaps between consonants: After arranging consonants, there are 6 gaps (e.g., _ S _ L _ C _ T _ N _). Place the vowels in these gaps, ensuring no gap has more than one vowel.
- Select 4 gaps from 6 to place vowels: .
- Arrange the vowels E, E, I, O in the selected gaps, considering repetition of E:
- Total arrangements:
(iv) All consonants are together
Treat all consonants (S, L, C, T, N) as a single "block." This block will be arranged along with the vowels (E, E, I, O), making a total of 5 items.
- Arrange the block + vowels:
- Within the consonant block: The 5 consonants can be arranged among themselves:
- Total arrangements:
Final Results:
- No restriction: 181,440
- Begin and end with "E": 5,040
- All vowels separated: 21,600
- All consonants together: 7,200
Would you like further clarifications on any part of the solution?
Here are 5 related questions you might explore:
- How would the results change if there were more than two repeating letters?
- What if all vowels must appear together instead of being separated?
- How does the inclusion of more restrictions impact the number of arrangements?
- Can you visualize the arrangement of gaps and letters for the vowel separation case?
- What if one of the consonants repeats as well?
Tip: Breaking problems into smaller logical steps often simplifies complex combinatorial calculations!
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Math Problem Analysis
Mathematical Concepts
Permutations
Combinatorics
Factorial Calculations
Formulas
Total permutations with repetitions = n! / (p1! * p2! * ...)
Permutations with restrictions, such as fixed positions or gap placements
Theorems
Fundamental principle of counting
Suitable Grade Level
Grades 10-12
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