What is the probability that a five-digit number formed by randomly arranging the digits 1, 2, 3, 4, and 5 is divisible by 4?

Instruction: Consider the divisibility rule for 4 and permutations.

Context: The question challenges the candidate's ability to combine knowledge of divisibility rules with combinatorial counting techniques.

Updated

Example Answer

If all arrangements are equally likely and each digit is used once, the probability is 1/5, or 20%. A number is divisible by four exactly when its final two digits form a multiple of four. Using distinct digits from one through five, the valid endings are 12, 24, 32, and 52.

For each ending, the other three digits can be arranged in 3! = 6 ways. That gives 4 × 6 = 24 qualifying numbers out of 5! = 120 possible arrangements.

Why this works

Writing a number as 100k + d shows why only its last two digits matter: 100k is divisible by four. As a second count, there are 5 × 4 = 20 equally likely ordered endings, and four qualify, again giving 1/5.

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