<p>Find the number of arrangements of all digits of 12345 such that at least 3 digits will not come in its positions.</p>
Step-by-Step Solution
Key Concept: Use inclusion-exclusion principle: Total arrangements minus arrangements where at least k digits ARE in correct positions. The condition 'at least 3 digits will NOT come in its positions' means at most 2 digits can be in correct positions.
<p><strong>Step 1:</strong> Interpret the condition. 'At least 3 digits will NOT come in its positions' means at most 2 digits can be in their correct positions. So we need: Arrangements where 0, 1, or 2 digits are in correct positions.</p><p><strong>Step 2:</strong> Use complementary counting: Total - (Arrangements with ≥3 digits in correct positions).</p><p><strong>Step 3:</strong> Calculate arrangements with exactly k digits in correct positions using inclusion-exclusion:</p><p>Exactly 3 correct: C(5,3) × D(2) = 10 × 1 = 10</p><p>Exactly 4 correct: C(5,4) × D(1) = 5 × 0 = 0</p><p>Exactly 5 correct: C(5,5) × D(0) = 1 × 1 = 1</p><p>Where D(n) is the derangement: D(2)=1, D(1)=0, D(0)=1</p><p><strong>Step 4:</strong> Arrangements with ≥3 digits in correct positions = 10 + 0 + 1 = 11</p><p><strong>Step 5:</strong> Total arrangements = 5! = 120</p><p>∴ Answer: 120 - 11 = <strong>109</strong></p>
Correct Answer: 109