Permutations & Combinations
Derangements
Grade 11
Question:
<p>Ajay writes letters to his five friends and addresses the corresponding envelopes. The number of ways the letters can be placed in the envelopes so that at least two of them are in the wrong envelopes is</p>
<p>(a) 120</p>
<p>(b) 125</p>
<p>(c) 119</p>
<p>(d) 124</p>
Step-by-Step Solution
Key Concept: Use the principle of derangements combined with complementary counting to find arrangements where at least two letters are in wrong envelopes.
<p><strong>Step 1:</strong> We need to find the number of ways letters can be placed so that at least two are in wrong envelopes.</p><p><strong>Step 2:</strong> This means we exclude the case where exactly one letter is in the correct envelope (which is impossible) and the case where all letters are in correct envelopes (which is 1 way).</p><p><strong>Step 3:</strong> Total arrangements = $5!$ = 120</p><p><strong>Step 4:</strong> Number of arrangements where at least 2 are in wrong envelopes = $\sum_{r=2}^{5} \binom{5}{5-r} D(r)$</p><p><strong>Step 5:</strong> Using derangement formula: $D(r) = r! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \ldots + \frac{(-1)^r}{r!}\right)$</p><p>Required number = 119</p><p>∴ Answer is (c) 119.</p>
Correct Answer: C