Question

A four-digit number is formed by the digits $1,2,3,4$ with no repetition.

The probability that the number is odd is

• A. Zero
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Find the probability that the number is odd:

Given: Digits $=\{1,2,3,4\}$

The event is the probability that the number is odd.

The unit digit of the number determines whether the number is even or odd.

If the unit digit of the number is $1$ then the possible way of arranging the remaining letters is $3!$ and the first three positions are arranged by remaining letters.

If the unit digit of the number is $3$ then the possible way of arranging the remaining letters is $3!$ and the first three positions are filled by remaining digits.

The event $E$ is the arrangement given by $3!+3!=12$ ways in which odd numbers are formed with digits $1,2,3$ and $4$.

$\Rightarrow n\left(E\right)=12$

The sample space is the number of numbers formed by $1,2,3,4$ without repetition is $4!=24$ ways.

$\Rightarrow n\left(S\right)=24$

Thus the probability of getting an odd number without repetition, $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$

$$\begin{gathered} =\frac{12}{24} \\ =\frac{\mathbf{1}}{\mathbf{2}} \end{gathered}$$

This option is not mentioned.

Hence option (D) is the correct option.