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. $\dfrac{1}{3}$
C. $\dfrac{1}{4}$
D. None of these

Answer 1

Hint: Let us find the total number of numbers whose last digit is odd because number is odd only and only if the last digit of the number is odd.

Complete step-by-step answer:
Let us find the total numbers that can be formed with digits 1, 2, 3, 4 with no repetition.
So, if no digit is repeated then there can be four digits available for first place.
3 digits available for second place.
2 digits available for third place.
And, 1 digit available for last or fourth place.
So, total numbers that are possible with given four digits without any repetition will be equal to 4321 = 24.
Now we had to find the total number out of 24 that are odd.
As we know that a number is odd only if its last digit is odd. So, here out of four given digits two are odd and that were 3 and 1.
So, first place can have two numbers.
And all three places can be filled with the two even numbers and one left odd number.
So, 2 digits (1,3) are available for first place.
3 digits (2,4 and anyone out of ${1,3}$ ) are available for second place.
2 digit is available for third place.
And ,1 digit is available for fourth place.
So, total numbers that can be formed with the given four digits without any repetition and are odd will be equal to 2321 = 12.
As we know that probability of a getting a favourable outcome is given as $\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total numbers of outcomes}}}}$.
Here favourable outcome is odd number and total outcome is all numbers.
So, probability = $\dfrac{{12}}{{24}}$ = $\dfrac{1}{2}$.
Hence, the correct option will be D.

Note: Whenever we come up with this type of problem then first, we find the total number of numbers that can be formed with four digits without any repetition. And then we will find a number of digits out of them that were odd. After that we had to apply a probability formula to get the required answer. And this will be the efficient way to find the solution of the problem.