Question

How many numbers greater than $24000$ can be formed by using digits $1,2,3,4,5$ when no digits are repeated?
$1)36$
$2)60$
$3)84$
$4)120$

Answer 1

First, we shall analyze the given data so that it can be easy for us to solve the problem. We are given five-unit digits $1,2,3,4,5$ that can be used to make the numbers that are greater than $24000$ . Follow the given step-by-step solution to solve the problem.
If once the digit is used then it will not be repeated again.

Complete step by step answer:
The given digits are $1,2,3,4,5$
Since we need to find the numbers that are greater than $24000$ , the first digit cannot be the digit $1$ (which is less than the given number)
If the digit in the ten thousand places is $2$ :
The number of possible digits for ten thousand places is $1$ (that is a digit $2$ )
The number of possible digits for thousands of place is $2$ (they are digits $4,5$ )
The number of possible digits for hundred places is $3$ (they are any digits which are not used)
The number of possible digits for tens place is $2$ (they are any digits which are not used)
The number of possible digits for one’s place is $1$ (they are any digits which is not used)
Hence, the total possible combination is $1 \times 2 \times 3 \times 2 \times 1 = 12$
If the digit in the ten thousand places is $3,4,5$ :
The number of possible digits for ten thousand places is $3$ (that is a digit $3(or)4(or)5$ )
The number of possible digits for thousands of place is $4$ (they are any digits which are not used)
The number of possible digits for hundred places is $3$ (they are any digits which are not used)
The number of possible digits for tens place is $2$ (they are any digits which are not used)
The number of possible digits for one’s place is $1$ (they are any digits which is not used)
Hence, the total possible combination is $3 \times 4 \times 3 \times 2 \times 1 = 72$
Therefore, combining the values we get, $12 + 72 = 84$

So, the correct answer is “Option C”.

Note: For constructing the five-digit number, the first digit cannot be the digit $1$ since we need to find the numbers that are greater than $24000$
Since repetition is not allowed, if we use the number $2$ then this same number will not be repeated in the process.