Identifying and Counting Numbers That Contain a Specific Digit Pattern

From a simple problem like finding 7 numbers that add up to 17, we can delve into the complexity of counting numbers containing a specific digit pattern. This exploration will not only enrich your understanding but also help optimize your content for Google search engines.

Introduction

In the realm of mathematics, one often encounters problems that at first glance might seem straightforward but, when examined closely, can reveal a depth of complexity. A classic example is finding 7 numbers that add up to 17. The simple solution of 1, 1, 1, 1, 1, 1, and 11 provides an answer. However, we can take this problem a step further by counting how many 7-digit numbers contain the pattern "17".

The Problem of Counting Numbers with a Specific Pattern

Let's explore this problem more generally for a base base b. We denote P1 as the count of permutations containing "17" exactly once, P2 as the count of permutations containing "17" exactly twice, and P3 as the count of permutations containing "17" exactly three times. The goal is to find the total count as P1P2P3.

Counting Permutations for a General Base

First, let's calculate the number of ways to embed "17" in a 7-digit number. Since the first digit cannot be zero, there are b5(b-1)b4 ways to do this. This method will count each permutation containing "17" once, twice, and thrice, respectively. Therefore, we have:

P12P23P3 6b5-5b4

Counting Specific Permutations

Next, we will count the number of ways to embed "17" twice in a 7-digit number. There are 4b3(6b-1)b2 ways to do this, which will count each permutation containing "17" twice once. We follow similar logic for further permutations.

Algorithm Implementation

Now, let's translate the mathematical concepts into a practical algorithm. Here's a simple C program to count the number of 7-digit numbers that contain the pattern "17".

include iostreaminclude stringusing namespace std;int main() {    int tot  0;    for(int i1000000; i10000000; i  ) {        string is  to_string(i);        int index  ("17");        if (index ! string::npos) {            tot  ;        }    }    cout  tot  endl;    return 0;}

This program iterates through all 7-digit numbers, converting each number to a string and checking if "17" is a substring. If "17" is found, the number is counted.

Dynamic Programming Approach

We can also solve this problem using dynamic programming. Let's define an as the count of n-digit numbers not containing "17" and ending with 1, and bn as the count of n-digit numbers not containing "17" and ending with a digit other than 1. Then, we have:

begin{bmatrix}a_1  b_1end{bmatrix}  begin{bmatrix}1  8end{bmatrix}begin{bmatrix}a_{n1}  b_{n1}end{bmatrix}  begin{bmatrix}1  1  8  9end{bmatrix}begin{bmatrix}a_n  b_nend{bmatrix}

By applying this recursively, we get:

begin{bmatrix}a_7  b_7end{bmatrix}  begin{bmatrix}1  1  8  9end{bmatrix}^6 begin{bmatrix}1  8end{bmatrix}  begin{bmatrix}854569  7604792end{bmatrix}

This results in the total count of 7-digit numbers not containing "17" as 9 * 106 - a7 - b7 540639.

To verify, we can run the Python function:

len(list(filter(lambda x: "17" in str(x), range(10**6, 10**7))))

This function filters out 7-digit numbers containing "17" and calculates the length of the resulting list, which is 540639.

Final Count: 540,639

Thus, the number of 7-digit numbers that contain the pattern "17" is 540,639. This solution not only satisfies the curiosity of mathematicians but also demonstrates the power of algorithmic thinking in solving complex problems.

Conclusion

From a simple mathematical problem, we have explored the broader implications of counting numbers with specific patterns and implemented various approaches to solve this problem. By understanding these concepts, SEO experts can create more comprehensive content that ranks well on Google search.

Keywords

digit pattern, permutations, number of combinations, Google SEO