Counting the Numbers

To determine the number of numbers greater than 24000 that can be formed using the digits 1, 2, 3, 4, and 5 without repetition, we need to consider the possible arrangements of these digits.

Step 1: Finding the Total Number of Arrangements
The total number of arrangements that can be formed using all the given digits without repetition is given by the formula for permutations of a set:

nPr = n! / (n-r)!

In this case, we have 5 digits (n = 5) and we need to choose 5 digits (r = 5) without repetition. Therefore, the total number of arrangements is:

5P5 = 5! / (5-5)! = 5! / 0! = 5! = 5 x 4 x 3 x 2 x 1 = 120

Step 2: Counting the Numbers Greater than 24000
To count the numbers greater than 24000, we need to consider the position of the first digit. Since the first digit must be greater than 2, there are only two options: 3 or 4.

Case 1: First Digit is 3
If the first digit is 3, then we have 4 remaining digits (1, 2, 4, 5) to choose from for the remaining 4 positions. The number of arrangements in this case is:

4P4 = 4! / (4-4)! = 4! / 0! = 4! = 4 x 3 x 2 x 1 = 24

Case 2: First Digit is 4
If the first digit is 4, then we have 3 remaining digits (1, 2, 5) to choose from for the remaining 4 positions. The number of arrangements in this case is:

3P4 = 3! / (4-4)! = 3! / 0! = 3! = 3 x 2 x 1 = 6

Total Number of Numbers Greater than 24000
To get the total number of numbers greater than 24000, we sum up the number of arrangements from both cases:

24 + 6 = 30

Therefore, there are 30 numbers greater than 24000 that can be formed using the digits 1, 2, 3, 4, and 5 without repetition.

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