how many words can be formed each of 2 vowels and 3 consonants from the letters of the given word - DAUGHTER

Let's break down the problem step-by-step.

### Given Word:
The word is **"DAUGHTER"**.

### Identifying Vowels and Consonants:
- **Vowels** in "DAUGHTER": A, U, E (3 vowels)
- **Consonants** in "DAUGHTER": D, G, H, T, R (5 consonants)

### Choosing the Letters:
1. **Choose 2 vowels** out of 3:  
   The number of ways to select 2 vowels from 3 is given by the combination formula:

   \[
   \binom{3}{2} = \frac{3!}{2!(3-2)!} = 3
   \]

2. **Choose 3 consonants** out of 5:  
   The number of ways to select 3 consonants from 5 is:

   \[
   \binom{5}{3} = \frac{5!}{3!(5-3)!} = 10
   \]

### Forming the Words:
Each selection of 2 vowels and 3 consonants gives us a set of 5 distinct letters. The number of ways to arrange 5 distinct letters is:

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Calculating the Total Number of Words:
The total number of words formed is given by multiplying the number of ways to choose the vowels and consonants, and the number of permutations of the selected letters:

\[
\binom{3}{2} \times \binom{5}{3} \times 5! = 3 \times 10 \times 120 = 3600
\]

Thus, the total number of words that can be formed is:

\[
\boxed{3600}
\]

Would you like me to explain any specific step further, or have any questions?

### Related Questions:
1. How many distinct permutations can be made from all the letters in "DAUGHTER"?
2. What is the number of words formed if 3 vowels and 2 consonants are chosen instead?
3. How many distinct 4-letter words can be formed from "DAUGHTER"?
4. What is the probability of randomly choosing a 5-letter word with at least 2 vowels from "DAUGHTER"?
5. If repetitions are allowed, how many 5-letter combinations can be made using "DAUGHTER"?

**Tip**: Always identify distinct groups (vowels and consonants) separately when dealing with letter arrangements in words.

Learn how to calculate the number of words formed by selecting 2 vowels and 3 consonants from the word 'DAUGHTER'. This combinatorics problem involves the use of combinations and permutations, making it ideal for students in Grades 9-12.

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