Permutations & Combinations
Word formation/Arrangements
Grade 11
Question:
<p>In how many ways can a word of five letters beginning with a capital letter and consisting of two vowels and two consonants be formed with the capital letters C, A, C; vowels o, u, i, e and the consonants b, l, p, t, s?</p>
Step-by-Step Solution
Key Concept: The first position must be filled with a capital letter (C or A), then arrange remaining 4 positions with exactly 2 vowels and 2 consonants selected from given sets. Order matters since we're forming distinct words.
<p><strong>Step 1: Fill the first position with a capital letter</strong></p><p>Available capitals: C, A, C (where C repeats)</p><p>Distinct choices: {C, A} → 2 ways</p><p><strong>Step 2: Select and arrange 2 vowels from {o, u, i, e}</strong></p><p>Number of ways = P(4,2) = 4!/(4-2)! = 4 × 3 = 12 ways</p><p><strong>Step 3: Select and arrange 2 consonants from {b, l, p, t, s}</strong></p><p>Number of ways = P(5,2) = 5!/(5-2)! = 5 × 4 = 20 ways</p><p><strong>Step 4: Arrange the 4 letters (2 vowels + 2 consonants) in remaining 4 positions</strong></p><p>Number of ways = 4! = 24 ways</p><p><strong>Step 5: Apply multiplication principle</strong></p><p>Total = 2 × 12 × 20 × 24 = 2 × 12 × 20 × 24 = 11,520 ways</p><p>∴ Answer: <strong>11,520</strong></p>
Correct Answer: 11