To solve this problem, we are tasked with finding how many different arrangements can be made from the letters of the word "COMBINE" under the condition that the vowels must appear in the same order.

Step 1: Analyze the word

The word "COMBINE" consists of 7 letters. The vowels are O, I, E (in that order), and the consonants are C, M, B, N.

Step 2: Consider the vowel condition

Since the vowels (O, I, E) must always appear in the same order, we can treat them as a single "block" or "unit." This means we are now arranging 5 items:

• The block of vowels (O, I, E)
• The consonants: C, M, B, N

Step 3: Permutations of the consonants and vowel block

We now have 5 "letters" to arrange: the vowel block (O, I, E) and the consonants (C, M, B, N). The number of ways to arrange these 5 items is simply the number of permutations of 5 distinct objects, which is given by:

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Final Answer:

Thus, the number of different arrangements of the word "COMBINE" where the vowels appear in the same order is 120.

Would you like more details on this, or do you have any further questions?

Related Questions:

1. How many ways can you arrange the letters of "COMBINE" if no restrictions are applied?
2. If the vowels must be next to each other but can appear in any order, how many arrangements are possible?
3. How many ways can you arrange "COMBINE" if the vowels must all be separated by consonants?
4. How many arrangements can be made if no two vowels can be next to each other in "COMBINE"?
5. What is the probability that a random arrangement of "COMBINE" will have vowels in alphabetical order?

Tip:

When dealing with letter arrangements and constraints, treat blocks of letters as single units to simplify the problem.