Permutations & Combinations
Derangements
Grade 11
Question:
<p>A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters.</p>
<p>(A) 20</p>
<p>(B) 40</p>
<p>(C) 60</p>
<p>(D) 15</p>
Step-by-Step Solution
Key Concept: Use combinations to select which letters go to correct houses, then use derangement formula for remaining letters.
<p><strong>Solution:</strong> To find the number of ways exactly 2 houses receive correct letters:</p><p>Step 1: Choose which 2 houses get correct letters: \(\binom{5}{2} = 10\) ways</p><p>Step 2: The remaining 3 letters must go to 3 houses such that none gets the correct letter (derangement of 3 objects): \(D_3 = 2\)</p><p>Step 3: Total ways = \(\binom{5}{2} \times D_3 = 10 \times 2 = 20\)</p><p>∴ Answer is (A) 20</p>
Correct Answer: A