Four digit numbers are formed by using the digits 1, 2, 3, 6 without repetition. The probability that a selected number is divisible by 3 is

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. 1

Answer

To solve the problem, we need to find the probability that a randomly selected four-digit number formed using the digits 1, 2, 3, and 6 without repetition is divisible by 3.

### Step-by-Step Solution:

1. **Identify the Total Number of Four-Digit Numbers:**
We can form a four-digit number using the digits 1, 2, 3, and 6. Since we are using all four digits without repetition, the total number of different four-digit numbers is given by the factorial of the number of digits:
\[ \text{Total Numbers} = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

2. **Check for Divisibility by 3:**
A number is divisible by 3 if the sum of its digits is divisible by 3. We need to calculate the sum of the digits we are using:
\[ \text{Sum of Digits} = 1 + 2 + 3 + 6 = 12 \]
Since 12 is divisible by 3, any arrangement of these digits will also yield a number that is divisible by 3.

3. **Determine Favorable Outcomes:**
Since all arrangements of the digits 1, 2, 3, and 6 will result in a number that is divisible by 3, the number of favorable outcomes is equal to the total number of four-digit numbers:
\[ \text{Favorable Outcomes} = 24 \]

4. **Calculate the Probability:**
The probability \( P \) that a randomly selected four-digit number is divisible by 3 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{24}{24} = 1 \]

### Final Answer:
The probability that a selected number is divisible by 3 is \( \boxed{1} \).