<p>The number of four-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 in which at least two digits are identical is</p>
Step-by-Step Solution
Key Concept: Use complementary counting: subtract four-digit numbers with all distinct digits from the total four-digit numbers formed with repetition allowed from the set {1,2,3,4,5}.
<p><strong>Step 1:</strong> Find the total number of four-digit numbers using digits {1, 2, 3, 4, 5} with repetition allowed.</p><p>Each of the 4 positions can be filled in 5 ways.</p><p>Total = 5⁴ = 625</p><p><strong>Step 2:</strong> Find the number of four-digit numbers with all four digits distinct.</p><p>Choose and arrange 4 different digits from 5 available digits.</p><p>All distinct = P(5,4) = 5!/(5-4)! = 5!/1! = 120</p><p><strong>Step 3:</strong> Apply complementary counting.</p><p>Numbers with at least two identical digits = Total - All distinct</p><p>= 625 - 120 = 505</p><p>∴ Answer: B</p>
Correct Answer: B