Question 32 of 42intermediate🔧 ApplyNumerical3 marks

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

Correct Answer

3600

Exercise: Miscellaneous Exercise on Chapter 6 | Q: 1 | (Page 23)
For More Understanding

Explanation

This question is from Miscellaneous Exercise Q1 and follows the same method as Example 20 in the textbook. The word DAUGHTER has 3 vowels (A, U, E) and 5 consonants (D, G, H, T, R). We need to select 2 vowels and 3 consonants, then arrange them. The solution involves combinations for selection and permutations for arrangement.

Solution Steps

  1. Step 1: Identify vowels and consonants in DAUGHTER. Vowels = {A, U, E} (3 vowels). Consonants = {D, G, H, T, R} (5 consonants).

  2. Step 2: Number of ways to select 2 vowels from 3 = ³C₂ = 3!/(2! × 1!) = 3 ways.

  3. Step 3: Number of ways to select 3 consonants from 5 = ⁵C₃ = 5!/(3! × 2!) = (5×45 \times 4)/2 = 10 ways.

  4. Step 4: Number of combinations of 2 vowels and 3 consonants = 3×103 \times 10 = 30 combinations.

  5. Step 5: Each combination has 5 letters which can be arranged in 5! ways = 120 ways.

  6. Step 6: Total number of words = 30×12030 \times 120 = 3600 words.

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