To find the probability that a three-digit number formed using the digits 1, 2, 3, and 4 without repetition is divisible by 3, follow these steps:
Step 1: Total number of three-digit numbers without repetition
- We have 4 digits: 1, 2, 3, 4.
- Number of ways to form a 3-digit number without repetition = permutations of 4 digits taken 3 at a time = P(4,3)=4×3×2=24P(4,3)=4\times 3\times 2=24P(4,3)=4×3×2=24.
Step 2: Divisibility rule for 3
- A number is divisible by 3 if the sum of its digits is divisible by 3
Step 3: Find all 3-digit combinations and their sums
We consider all 3-digit numbers formed from {1, 2, 3, 4} without repetition and check the sum of digits:
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Possible 3-digit digit sets (without order):
(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4) -
Sum of digits for each set:
- 1 + 2 + 3 = 6 (divisible by 3)
- 1 + 2 + 4 = 7 (not divisible by 3)
- 1 + 3 + 4 = 8 (not divisible by 3)
- 2 + 3 + 4 = 9 (divisible by 3)
Only the sets (1, 2, 3) and (2, 3, 4) have sums divisible by 3.
Step 4: Count favorable numbers
- For each set of 3 digits, the number of permutations (3-digit numbers) = 3! = 6.
- Sets with sum divisible by 3: 2 sets × 6 permutations each = 12 favorable numbers.
Step 5: Calculate probability
Probability=Number of favorable outcomesTotal number of outcomes=1224=12\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{12}{24}=\frac{1}{2}Probability=Total number of outcomesNumber of favorable outcomes=2412=21
Final answer: The probability that the number formed is divisible by 3 is 12\boxed{\frac{1}{2}}21. This matches the answer choice (C) 1/2 from the related source
