Question

If five-digit numbers are formed from the digits 0, 1, 2, 3, 4 using every digit exactly only once, then the probability that a randomly chosen number from those numbers is divisible by 4 is

• A. \( \frac{5}{16} \)
• B. \( \frac{3}{16} \)
• C. \( \frac{3}{8} \)
• D. \( \frac{7}{16} \)

Hint:

When dealing with permutations and probability, carefully consider the restrictions placed on digit positions, especially for divisibility rules.

Answer 1

The Correct Option is A

Step 1: Understand the condition for divisibility by 4. A number is divisible by 4 if its last two digits form a number that is divisible by 4. Therefore, we only need to focus on the last two digits of the five-digit number. 
Step 2: Count the total number of possible five-digit numbers that can be formed using each of the digits 0, 1, 2, 3, 4 exactly once. - There are 5 digits in total, but the first digit cannot be 0 (since it's a five-digit number). - Therefore, the total number of five-digit numbers is \( 4! = 24 \). 
Step 3: Determine the valid combinations of the last two digits (12, 24, 32, and 40) that make the number divisible by 4. - A number formed by the digits 0, 1, 2, 3, 4 is divisible by 4 if its last two digits are divisible by 4. - The valid combinations of the last two digits that satisfy this condition are: - 12 - 24 - 32 - 40 
Step 4: For each valid pair, calculate the number of permutations of the remaining three digits. - After choosing the last two digits, the remaining three digits can be arranged in \( 3! = 6 \) ways. - Hence, for each valid pair of the last two digits, there are 6 possible numbers. 
Step 5: Add the total number of permissible five-digit numbers. - There are 4 valid pairs for the last two digits (12, 24, 32, and 40), and for each pair, we have 6 valid numbers. - Therefore, the total number of valid five-digit numbers is \( 4 \times 6 = 24 \). 
Step 6: Calculate the probability. - The probability that a randomly chosen five-digit number is divisible by 4 is the ratio of valid numbers to the total number of numbers: \[ P(\text{divisible by 4}) = \frac{\text{Number of valid combinations}}{\text{Total number of numbers}} = \frac{4 \times 6}{4!} = \frac{24}{24} = 1. \]