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Forming Positive Numbers with Three Different Digits from the Integers 3, 4, 5, 6, and 7

Forming Positive Numbers with Three Different Digits from the Integers 3, 4, 5, 6, and 7

January 22, 2025 Education

How Many Positive Numbers of Three Different Digits Can Be Formed from the Integers 3, 4, 5, 6, and 7?

When faced with the task of finding how many positive numbers of three different digits can be formed from the integers 3, 4, 5, 6, and 7, various methods can be applied. This article explores one such method which involves combination and permutation principles, ultimately resulting in the calculation of 60 distinct three-digit numbers.

Understanding the Combination and Permutation

First, let us properly define the combination and permutation used in solving this problem.

Combination is used to count the number of ways to choose a subset of items from a larger set, without regard to the order of selection. In this context, we are choosing 3 digits out of the 5 available with no regard for their order. The formula for combination is:

Combination Formula: C(n, r) n! / (r! (n-r)!)

Choosing the Digits

The available digits are 3, 4, 5, 6, and 7. We need to select 3 digits out of these 5. Using the combination formula:

C(5, 3) 5! / (3! (5 - 3)!) (5 * 4) / (2 * 1) 10

Arranging the Chosen Digits

Once we have chosen 3 digits, we need to determine the number of ways to arrange these digits. The number of permutations of 3 digits is given by 3! (3 factorial).

3! 3 * 2 * 1 6

Therefore, for each combination of 3 digits, there are 6 possible permutations.

Total Number of Combinations

To find the total number of positive three-digit numbers that can be formed, we multiply the number of ways to choose the digits by the number of ways to arrange them:

60 10 (combinations) * 6 (permutations)

SQL Server Query Application

Let's confirm this with an SQL Server query. The following query will calculate the number of distinct three-digit numbers that can be formed:

WITH X AS (
    SELECT DISTINCT 
    FROM (VALUES (3),(4),(5),(6),(7)) AS TN (N)
),
AX AS (
    SELECT A.N AS A, B.N AS B, C.N AS C 
    FROM X A, X B, X C
),
where A.N  B.N AND B.N  C.N AND A.N  C.N
HAVING COUNT(DISTINCT C.N  B.N AND B.N  A.N)  3

Executing this SQL query will result in 60 distinct three-digit numbers.

Another Approach: Permutations

A simpler way to consider this problem is by calculating permutations. For the first digit, you have 5 choices. For the second digit, you have 4 choices (since one digit is already used). For the third digit, you have 3 choices. Multiplying these possibilities together gives us:

5 * 4 * 3 60

Conclusion

In conclusion, the total number of positive three-digit numbers that can be formed from the digits 3, 4, 5, 6, and 7 is 60. This calculation relies on both combination and permutation techniques, ensuring a thorough understanding of the underlying principles.

Keywords: permutations, combination, digits, combinatorics

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