Permutations & Combinations
Derangements
Grade 11
Question:
<p>Find the number of ways in which five different letters can be put in their five addressed envelopes so that all the letters are in the wrong envelopes.</p>
Step-by-Step Solution
Key Concept: This is a derangement problem where we need to find the number of permutations of 5 objects such that no object is in its original position. Use the derangement formula: D(n) = n![1 - 1/1! + 1/2! - 1/3! + ... + (-1)ⁿ/n!]
<p><strong>Step 1:</strong> Recognize this as a derangement problem. We need all 5 letters in wrong envelopes (no letter in its correct position).</p><p><strong>Step 2:</strong> Apply the derangement formula: D(n) = n! × Σ((-1)^k / k!) for k = 0 to n</p><p><strong>Step 3:</strong> Calculate D(5):</p><p>D(5) = 5![1/0! - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!]</p><p>= 120[1 - 1 + 1/2 - 1/6 + 1/24 - 1/120]</p><p>= 120[0 + 1/2 - 1/6 + 1/24 - 1/120]</p><p><strong>Step 4:</strong> Find common denominator (LCD = 120):</p><p>= 120[(60 - 20 + 5 - 1)/120]</p><p>= 120 × 44/120</p><p>= 44</p><p><strong>∴ Answer: 44</strong></p>
Correct Answer: 44