How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ?

Options

A. 30
B. 3000
C. 36
D. 3600

Explanation

To solve the problem of how many words can be formed using 2 vowels and 3 consonants from the letters of the word "DAUGHTER," we can follow these steps:

### Step 1: Identify the Vowels and Consonants
The letters in the word "DAUGHTER" are:
- Vowels: A, U, E (3 vowels)
- Consonants: D, G, H, T, R (5 consonants)

### Step 2: Choose 2 Vowels from 3
We need to select 2 vowels from the 3 available vowels (A, U, E). The number of ways to choose 2 vowels from 3 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 vowels} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]

### Step 3: Choose 3 Consonants from 5
Next, we need to select 3 consonants from the 5 available consonants (D, G, H, T, R). The number of ways to choose 3 consonants from 5 can be calculated using the combination formula:

\[ \text{Number of ways to choose 3 consonants} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Calculate Total Combinations of Vowels and Consonants
Now, we multiply the number of ways to choose the vowels by the number of ways to choose the consonants:

\[ \text{Total combinations} = \binom{3}{2} \times \binom{5}{3} = 3 \times 10 = 30 \]

### Step 5: Arrange the Selected Letters
Each combination of 2 vowels and 3 consonants consists of 5 different letters. The number of ways to arrange 5 different letters is given by the factorial of 5:

\[ \text{Number of arrangements} = 5! = 120 \]

### Step 6: Calculate Total Words Formed
Finally, we multiply the total combinations of letters by the number of arrangements:

\[ \text{Total words} = \text{Total combinations} \times \text{Number of arrangements} = 30 \times 120 = 3600 \]

### Conclusion
Thus, the total number of words (with or without meaning) that can be formed using 2 vowels and 3 consonants from the letters of the word "DAUGHTER" is **3600**.

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