Calculating Unique Seven-Digit Numbers from Given Digits

Permutations are a fundamental concept in combinatorics, the branch of mathematics dealing with the arrangement of objects in specific orders. This article will explore the problem of determining the number of unique seven-digit numbers that can be formed from the digits 9, 9, 2, 2, 5, 5, and 5. We will present the mathematical reasoning and computational methods behind this calculation, ensuring the results align with Google's search standards.

Introduction to Permutations

When all 7 digits are distinct, the number of permutations is simple: 7! 5040. However, since the digits 9, 2, and 5 are repeated, we need to adjust the calculation to account for these repeated digits. The formula to account for repetitions is given by n! / (n1! x n2! x nk!), where n1, n2, and nk are the counts of the repeating digits.

Adjusting for Repetitions

In this specific problem, we have two 9s, two 2s, and three 5s. The total number of unique permutations is calculated as:

7! / (2! x 2! x 3!) 5040 / 24 210

Alternatively, we can break down the process by considering the placement of the digits step-by-step:

7C3 ways to place the three 5s within the seven positions. 4C2 ways to place the two 2s within the remaining four positions. 2C2 ways to place the two 9s within the last two positions.

Multiplying these combinations together, we get:

7C3 x 4C2 x 2C2 210

Computational Approach

For a more concrete visualization, we can write a simple C program to generate and count these unique seven-digit numbers. Here is an example of such a program:

include stdio.h
int main() {
    char s[10];
    int i, j, t  0, n2, n5, n9;
    s[0]  '9'; s[1]  '9'; s[2]  '2'; s[3]  '2'; s[4]  '5'; s[5]  '5'; s[6]  '5';
    for (i  2255599; i  9955522; i  ) {
        sprintf(s, "%d%d%d%d%d%d%d", i / 1000000, i / 100000 % 10, i / 10000 % 10, i / 1000 % 10, i / 100 % 10, i / 10 % 10, i % 10);
        n2  0; n5  0; n9  0;
        for (j  0; j  7; j  ) {
            if (s[j]  '2') n2  ;
            if (s[j]  '5') n5  ;
            if (s[j]  '9') n9  ;
        }
        if (n2  2  n5  3  n9  2) {
            t  ;
        }
    }
    printf("Total unique 7-digit combinations: %d
", t);
}

This program checks every possible seven-digit number formed from the given digits and counts those that meet the specified criteria.

Conclusion

Understanding permutations and how to adjust for repetitions is crucial in many combinatorial problems. By applying the correct mathematical principles, we can determine that there are 210 unique seven-digit numbers that can be formed from the digits 9, 9, 2, 2, 5, 5, and 5. This result can be confirmed through both mathematical reasoning and computational verification.