Question

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

• A. \(120\)
• B. \(21600\)
• C. \(720\)
• D. \(3600\)

Hint:

For forming words using selected vowels and consonants, first choose the required letters and then arrange all selected letters using factorial.

Answer 1

The Correct Option is D

Step 1: List the letters in DAUGHTER.
The word DAUGHTER has 8 letters: D, A, U, G, H, T, E, R. We classify them as vowels and consonants.
Step 2: Identify vowels and consonants.
Vowels: A, U, E (3 vowels). Consonants: D, G, H, T, R (5 consonants).
Step 3: Choose 2 vowels from 3.
$ {}^3C_2 = 3 $ ways.
Step 4: Choose 3 consonants from 5.
$ {}^5C_3 = 10 $ ways.
Step 5: Arrange the 5 selected letters.
We have selected $ 2 + 3 = 5 $ letters. These can be arranged in $ 5! = 120 $ ways (each arrangement is a different word).
Step 6: Multiply to get the total.
Total = $ {}^3C_2 \times {}^5C_3 \times 5! = 3 \times 10 \times 120 = 3600 $. \[ \boxed{3600} \]