A four-digit number is formed by the digits 1,2,3,4 with no repetition. The probability that the number is odd, is

Options

A. zero
B. `(1)/(3)`
C. `(1)/(4)`
D. None of these

Explanation

To solve the problem of finding the probability that a four-digit number formed by the digits 1, 2, 3, and 4 (with no repetition) is odd, we can follow these steps:

### Step 1: Identify the total number of four-digit combinations
Since we are using the digits 1, 2, 3, and 4 without repetition, we can calculate the total number of four-digit combinations using the factorial of the number of digits available.

\[ \text{Total combinations} = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 2: Determine the criteria for the number to be odd
A number is odd if its last digit is an odd number. From the digits we have (1, 2, 3, 4), the odd digits are 1 and 3. Therefore, we have two choices for the last digit.

### Step 3: Calculate the number of favorable outcomes
1. **Case 1**: If the last digit is 1:
- The remaining digits are 2, 3, and 4.
- The number of ways to arrange these three digits is \(3!\):
\[ 3! = 3 \times 2 \times 1 = 6 \]

2. **Case 2**: If the last digit is 3:
- The remaining digits are 1, 2, and 4.
- The number of ways to arrange these three digits is also \(3!\):
\[ 3! = 3 \times 2 \times 1 = 6 \]

Adding both cases together gives us the total number of favorable outcomes:
\[ \text{Total favorable outcomes} = 6 + 6 = 12 \]

### Step 4: Calculate the probability
The probability \(P\) that a randomly formed four-digit number is odd is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{odd}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{12}{24} = \frac{1}{2} \]

### Conclusion
Thus, the probability that the four-digit number is odd is:
\[ \boxed{\frac{1}{2}} \]