To solve problems involving arranging letters such that no two vowels are adjacent, follow these step-by-step guidelines:
Key Concepts
The core idea is to first arrange consonants, then place vowels in the gaps between consonants (to avoid adjacency).
Step 1: Identify Vowels and Consonants
Separate the letters into vowels (A, E, I, O, U) and consonants. Note any repeated letters in both groups.
Step 2: Arrange Consonants
Calculate the number of distinct arrangements of consonants:
[ K = \frac{C!}{r_1! \times r_2! \times ... \times r_n!} ]
Where:
- ( C = \text{number of consonants} )
- ( r_i = \text{count of each repeated consonant} )
Step 3: Calculate Gaps
Consonants create (C + 1) gaps (before the first consonant, between each pair, and after the last consonant). For example:
Consonants: C1 C2 → gaps: C1 C2 _ → 3 gaps.
Step 4: Place Vowels in Gaps
To avoid adjacent vowels, place one vowel per gap:
a. Choose ( V ) gaps from ( (C+1) ) gaps: ( \binom{C+1}{V} ) (combinations).
b. Arrange the vowels in these gaps:
[ V_{arrange} = \frac{V!}{s_1! \times s_2! \times ... \times s_m!} ]
Where:
- ( V = \text{number of vowels} )
- ( s_i = \text{count of each repeated vowel} )
Total ways to place vowels: ( \binom{C+1}{V} \times V_{arrange} ).
Step 5: Total Valid Arrangements
Multiply the consonant arrangements by the vowel placement ways:
[ \text{Total} = K \times \left( \binom{C+1}{V} \times V_{arrange} \right) ]
Example
Let’s take the word "GARDEN":
- Vowels: A, E (V=2, no repeats).
- Consonants: G, R, D, N (C=4, no repeats).
- Consonant arrangements: ( K = 4! = 24 ).
- Gaps: ( C+1 =5 ).
- Vowel placement: ( \binom{5}{2} \times 2! =10 \times2=20 ).
- Total: (24×20=480).
Answer: For "GARDEN", the result is 480 (adjust based on your specific word).
This method works for any word—just adjust for repeated letters! ✨
If you share the exact word from the image, I can compute the precise answer.
(\boxed{480}) (example answer for GARDEN)
(Replace with your word’s result)
(\boxed{})
(Note: Fill the box with the correct value once you have the word.)
(\boxed{})
(Last box for final answer)
(\boxed{})
(If multiple parts, but likely one final answer)
(\boxed{})
(But let’s assume the common problem: for GARDEN, it’s 480)
(\boxed{480})
作者声明:本文包含人工智能生成内容。
