In how many ways can the letters of the word "RAINBOW" be arranged if the consonants must be in alphabetical order?

The word "RAINBOW" has 7 letters: R, A, I, N, B, O, and W.

To solve the problem, we are given that the **consonants must be in alphabetical order**. Let's break down the solution step by step.

### Step 1: Identify the consonants and vowels
- **Consonants:** R, N, B, W (4 consonants)
- **Vowels:** A, I, O (3 vowels)

### Step 2: Fix the consonants in alphabetical order
Since the consonants must be in alphabetical order, the consonants B, N, R, and W must appear in that order in any arrangement of the word. Their relative positions are fixed, and we cannot change their order.

Thus, the problem reduces to selecting 4 positions for the consonants from the 7 available positions in the word "RAINBOW".

### Step 3: Choose positions for the consonants
We need to choose 4 positions out of the 7 available for the consonants. The number of ways to do this is given by the combination formula:

\[
\binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

### Step 4: Place the vowels
After the consonants have been placed in the selected positions, the remaining 3 positions will automatically be filled by the vowels A, I, and O. These vowels can be arranged in any order, and the number of ways to arrange 3 vowels is given by:

\[
3! = 3 \times 2 \times 1 = 6
\]

### Step 5: Multiply the results
Now, to find the total number of possible arrangements, we multiply the number of ways to choose the positions for the consonants by the number of ways to arrange the vowels:

\[
\text{Total arrangements} = \binom{7}{4} \times 3! = 35 \times 6 = 210
\]

### Final Answer:
The number of ways the letters of the word "RAINBOW" can be arranged if the consonants must be in alphabetical order is **210**.

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Do you need any further clarifications or details?

Here are 5 related questions:
1. What if the consonants and vowels must alternate in the arrangement?
2. How many ways can the letters of "RAINBOW" be arranged if the vowels must be in alphabetical order?
3. How many ways can the letters of "RAINBOW" be arranged with no restrictions?
4. What if two letters in "RAINBOW" are indistinguishable? How would that affect the arrangement count?
5. Can you calculate the probability that a random arrangement of "RAINBOW" has its consonants in alphabetical order?

**Tip:** For combination problems, remember that ordering constraints on specific elements often reduce the total number of possibilities.

In how many ways can the letters of the word 'RAINBOW' be arranged if the consonants must be in alphabetical order?

Discover how to calculate the number of ways to arrange the letters of 'RAINBOW' when the consonants must remain in alphabetical order. This problem utilizes combinatorics concepts like combinations and permutations. Detailed solution included for high school students, grades 9-12.

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