To determine how many arrangements can be formed using all the letters in "combine" such that the vowels (O, I, E) must be placed in the 2nd, 5th, and 7th positions, follow these steps:

Step 1: Identify the vowels and consonants.

• The word "combine" has 7 letters in total.
• Vowels: O, I, E (3 vowels).
• Consonants: C, M, B, N (4 consonants).

Step 2: Assign the vowels to specific positions.

The vowels must be placed in the 2nd, 5th, and 7th positions. So, we will arrange the vowels in these specific spots. The number of ways to arrange 3 vowels in 3 positions is:

$3! = 3 \times 2 \times 1 = 6 \text{ ways}.$

Step 3: Arrange the consonants in the remaining positions.

The remaining positions are the 1st, 3rd, 4th, and 6th, where the consonants must be placed. The number of ways to arrange 4 consonants in 4 positions is:

$4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}.$

Step 4: Calculate the total number of arrangements.

To get the total number of arrangements, multiply the number of ways to arrange the vowels by the number of ways to arrange the consonants:

$6 \times 24 = 144 \text{ total arrangements}.$

Thus, the total number of arrangements where the vowels are in the 2nd, 5th, and 7th positions is 144.

Would you like more details or have any further questions?

Here are some related questions you might consider:

1. How does the number of arrangements change if one of the letters appears more than once?
2. What if the vowels could be placed in any position?
3. How many arrangements are there if the word has 8 letters instead of 7?
4. How many vowel arrangements are possible if another vowel is added to the set?
5. What if the consonants must also be in specific positions?

Tip: When arranging objects in specific positions, the factorial function is commonly used to calculate permutations.