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

Answer

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 will follow these steps:

### Step 1: Calculate the total number of four-digit numbers that can be formed.
Since we have 4 distinct digits (1, 2, 3, 4), the total number of different four-digit numbers that can be formed is given by the factorial of the number of digits:

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

**Hint:** Remember that the factorial of a number \( n \) (denoted \( n! \)) is the product of all positive integers up to \( n \).

### Step 2: Identify the criteria for the number to be odd.
A four-digit number is odd if its last digit is an odd number. In our case, the odd digits available are 1 and 3.

### Step 3: Calculate the number of odd four-digit numbers.
We will consider two cases based on the last digit being either 1 or 3.

#### Case 1: Last digit is 1
- If the last digit is fixed as 1, we can use the remaining digits (2, 3, 4) to fill the first three positions.
- The number of arrangements of the remaining three digits is given by \( 3! \):

\[ \text{Numbers with last digit 1} = 3! = 3 \times 2 \times 1 = 6 \]

#### Case 2: Last digit is 3
- If the last digit is fixed as 3, we can use the remaining digits (1, 2, 4) to fill the first three positions.
- Again, the number of arrangements of the remaining three digits is given by \( 3! \):

\[ \text{Numbers with last digit 3} = 3! = 3 \times 2 \times 1 = 6 \]

### Step 4: Calculate the total number of odd numbers.
Now, we can add the results from both cases:

\[ \text{Total odd numbers} = 6 + 6 = 12 \]

### Step 5: Calculate the probability that a number is odd.
The probability \( P \) that a randomly formed four-digit number is odd is given by the ratio of the number of odd numbers to the total number of four-digit numbers:

\[ P(\text{odd}) = \frac{\text{Total odd numbers}}{\text{Total numbers}} = \frac{12}{24} = \frac{1}{2} \]

### Conclusion
The probability that a four-digit number formed by the digits 1, 2, 3, and 4 (with no repetition) is odd is:

\[ \boxed{\frac{1}{2}} \]