<p>How many numbers can be formed from the digits 1, 2, 3, 4 when repetition is not allowed?</p>
Step-by-Step Solution
Key Concept: The question asks for ALL possible numbers (of any length from 1 to 4 digits), not just 4-digit numbers. Sum the permutations for 1-digit, 2-digit, 3-digit, and 4-digit numbers separately.
<p><strong>Step 1:</strong> Identify what needs to be counted. We can form 1-digit, 2-digit, 3-digit, and 4-digit numbers from {1, 2, 3, 4} without repetition.</p><p><strong>Step 2:</strong> Count 1-digit numbers: Choose 1 digit from 4 = <sup>4</sup>P₁ = 4</p><p><strong>Step 3:</strong> Count 2-digit numbers: Choose and arrange 2 digits from 4 = <sup>4</sup>P₂ = 4 × 3 = 12</p><p><strong>Step 4:</strong> Count 3-digit numbers: Choose and arrange 3 digits from 4 = <sup>4</sup>P₃ = 4 × 3 × 2 = 24</p><p><strong>Step 5:</strong> Count 4-digit numbers: Arrange all 4 digits = <sup>4</sup>P₄ = 4 × 3 × 2 × 1 = 24</p><p><strong>Step 6:</strong> Total = <sup>4</sup>P₁ + <sup>4</sup>P₂ + <sup>4</sup>P₃ + <sup>4</sup>P₄ = 4 + 12 + 24 + 24 = <strong>64</strong></p>
Correct Answer: \({}^{4}P_1 + {}^{4}P_2 + {}^{4}P_3 + {}^{4}P_4\)