To solve the problem of arranging the letters of the word "EDUCATION" under the given conditions, we will follow these steps:
### Step 1: Identify the vowels and consonants
The word "EDUCATION" consists of the following letters:
- Vowels: E, U, A, I, O (5 vowels)
- Consonants: D, C, T, N (4 consonants)
### Step 2: Determine the arrangement of vowels
According to the problem, the vowels must occur in the order E, U, A, I, O. Since the order is fixed, we do not need to arrange them further. The vowels will take specific positions in the arrangement.
### Step 3: Determine the arrangement of consonants
The consonants D, C, T, N must also occur in the order D, C, T, N. Similar to the vowels, since the order is fixed, we do not need to arrange them further.
### Step 4: Place vowels and consonants with the condition that no two consonants are adjacent
To satisfy the condition that no two consonants are next to each other, we can visualize the arrangement as follows:
1. First, we place the vowels in the arrangement. Since there are 5 vowels, they will create 6 gaps (including the ends) where consonants can be placed:
- _ V _ V _ V _ V _ V _
- (where V represents a vowel)
2. The 6 gaps are:
- Before the first vowel
- Between the first and second vowel
- Between the second and third vowel
- Between the third and fourth vowel
- Between the fourth and fifth vowel
- After the fifth vowel
### Step 5: Choose gaps for consonants
We need to choose 4 out of these 6 gaps to place the consonants D, C, T, N. The number of ways to choose 4 gaps from 6 is given by the combination formula:
\[
\binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15
\]
### Step 6: Calculate the total arrangements
Since both the vowels and consonants have fixed orders, the total number of arrangements is simply the number of ways to choose the gaps for the consonants, which we calculated in the previous step. Thus, the total number of arrangements is:
\[
15
\]
### Final Answer
The total number of ways to arrange the letters of the word "EDUCATION" under the given conditions is **15**.
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