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Let's break down both questions step by step.

### Question 3:
**In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?**

1. **Word Analysis:**
   - The word "CORPORATION" has 11 letters in total.
   - Vowels: O, O, A, I, O (5 vowels in total).
   - Consonants: C, R, P, R, T, N (6 consonants in total).

2. **Treating the vowels as a block:**
   If the vowels always come together, treat the 5 vowels as a single entity. Therefore, we are now arranging 6 consonants + 1 block of vowels, i.e., 7 entities.

3. **Arranging the 7 entities (consonants + vowel block):**
   The number of ways to arrange 7 entities is \( 7! \) (7 factorial).

4. **Arranging the vowels within the block:**
   The vowels within the block can be arranged among themselves. However, since the vowels "O" appears 3 times, the number of ways to arrange them is \( \frac{5!}{3!} \).

Thus, the total number of arrangements is:
\[
7! \times \frac{5!}{3!}
\]

Now let's calculate this:
- \( 7! = 5040 \)
- \( 5! = 120 \)
- \( 3! = 6 \)

So, the total number of ways is:
\[
5040 \times \frac{120}{6} = 5040 \times 20 = 100800
\]

There seems to be an issue with the provided options (since 100800 isn't listed), but based on the mathematical process, this is the correct solution.

### Question 4:
**Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?**

1. **Choosing consonants:**
   The number of ways to choose 3 consonants from 7 is \( \binom{7}{3} \).

2. **Choosing vowels:**
   The number of ways to choose 2 vowels from 4 is \( \binom{4}{2} \).

3. **Arranging the selected consonants and vowels:**
   Once 3 consonants and 2 vowels are chosen, they can be arranged in \( 5! \) ways (since the word will have 5 letters).

Thus, the total number of ways is:
\[
\binom{7}{3} \times \binom{4}{2} \times 5!
\]

Now let's calculate this:
- \( \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \)
- \( \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \)
- \( 5! = 120 \)

So, the total number of ways is:
\[
35 \times 6 \times 120 = 25200
\]

The correct answer is **25200**, which matches option (C).

---

Do you need further clarification or details on any part of this? Let me know!

### 5 Related Questions:
1. How would the answer to Question 3 change if only 4 vowels had to come together?
2. What if the vowels and consonants had to alternate in Question 3?
3. How many different ways can the letters of "CORPORATION" be arranged if there were no restrictions?
4. What if you had to form words using 2 consonants and 3 vowels from Question 4?
5. How does the presence of repeated letters (like "O" and "R") affect the total arrangements in these problems?

### Tip:
Factorial-based problems often involve simplifying repeated elements (like vowels or consonants) by dividing out repeated arrangements to avoid overcounting!

In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together? Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

Learn how to solve two combinatorics problems involving the arrangement of letters in the word 'CORPORATION' where vowels must always come together, and forming words with a specified number of consonants and vowels. Understand how to apply permutation, factorial, and combination concepts.

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