How Many Numbers Greater Than 2000 Can Be Formed With the Digits 1 2 3 4 and 5?
When addressing the problem of forming numbers greater than 2000 using specific digits, we can break it down into manageable parts. This article will guide you through the process of determining how many 4- and 5-digit numbers can be formed with the digits 1, 2, 3, 4, and 5, each used distinctly.
1. Count 4-Digit Numbers Greater Than 2000
A 4-digit number is greater than 2000 if its first digit (thousands place) is either 2, 3, 4, or 5. Let's explore the possibilities in each case:
First digit is 2: The remaining digits are 1, 3, 4, and 5. The number of ways to arrange these 3 digits is 3! 6 First digit is 3: The remaining digits are 1, 2, 4, and 5. The number of ways to arrange these 3 digits is 3! 6 First digit is 4: The remaining digits are 1, 2, 3, and 5. The number of ways to arrange these 3 digits is 3! 6 First digit is 5: The remaining digits are 1, 2, 3, and 4. The number of ways to arrange these 3 digits is 3! 6Total for 4-digit numbers: 6 6 6 6 24
2. Count 5-Digit Numbers
For 5-digit numbers, since the first digit (ten-thousands place) can be any of the available digits 1, 2, 3, 4, or 5, all such numbers will be greater than 2000. We need to find the number of ways to arrange all 5 digits.
The number of ways to arrange all 5 digits is 5! 120.
3. Final Calculation
We add the counts of 4-digit and 5-digit numbers to get the total:
Total: 24 (4-digit) 120 (5-digit) 144
Therefore, the total number of numbers greater than 2000 that can be formed with the digits 1, 2, 3, 4, and 5, each used distinctly, is 144.
Additional Insight
The problem statement also mentions an alternative way of counting the integers. Let's consider the following scenarios:
4-Digit Integers
A 4-digit integer is represented as WXYZ, where W is the Most Significant Digit (MSD). There are 4 choices for W {2, 3, 4, 5}, and then 4 choices for X (since 1 becomes available), 3 for Y, and 2 for Z. Thus, the total number of 4-digit integers is:
4 x 4 x 3 x 2 96
5-Digit Integers
A 5-digit integer is represented as VWXYZ, where V is the MSD. There are 5 choices for V {1, 2, 3, 4, 5}, 4 for W, 3 for X, 2 for Y, and 1 for Z. Thus, the total number of 5-digit integers is:
5 x 4 x 3 x 2 x 1 120
TOTAL 96 (4-digit) 120 (5-digit) 216 integers
By putting together the detailed analysis and the alternative counting methods, we have a comprehensive understanding of the problem and the answer.