Understanding and Counting Specific Bit Strings

Counting bit strings that meet specific criteria, such as containing a certain number of zeros (0s) and ones (1s) with particular rules, can be an intriguing problem in combinatorics and algorithm design. One such interesting problem involves counting bit strings that contain exactly three 0s and fifteen 1s, with each 0 being immediately followed by three 1s. This article delves into the problem and provides a detailed solution.

The Problem Restated

Given bit strings containing exactly three 0s and fifteen 1s, with each 0 being immediately followed by three 1s, the problem is to determine the total number of such bit strings. The solution to this problem is 84.

Understanding the Solution

One effective method to solve this problem involves transforming it into a simpler combinatorial problem by considering the 0s and 1s in groups. Specifically, each 0 and its trailing three 1s form a block of four bits (0111).

Block Formation

The problem can be rephrased as counting the number of ways to arrange five 0111 blocks and four additional 1s (the remaining 1s after the 0111 blocks are placed). This is a classic combinatorial problem of arranging two types of items: blocks and individual 1s. The number of such arrangements can be calculated using the combinations formula.

The number of ways to choose 4 positions out of 9 for the individual 1s is given by the binomial coefficient binom{9}{4} frac{9!}{5!4!} 126.

Alternative Combinatorial Approaches

Another combinatorial approach involves transforming the problem by replacing each 0 with a single symbol (g) representing "0111" and each individual 1 with another symbol (o). The problem then becomes arranging three "g" symbols and six "o" symbols in any order. The number of such arrangements is given by binom{9}{3} frac{9!}{3!6!} 84.

Algorithmic Solution

To implement an algorithm for generating and counting such bit strings, we can use a state machine approach. Here's a step-by-step guide:

State Machine Design

Initialize a state State 0, which means we are looking for 1s by themselves or 0111 sequences. Iterate through each bit in the string: If in State 0 and a 1 is seen, stay in State 0. If in State 0 and a 0 is seen, move to State 1 (looking for 111). If in State 1 and a 1 is seen, move to State 2 (111 found, so move back to 0). If in State 2 and a 0 is seen, this is a fail (invalid bit string). If in State 3 and a 1 is seen, move back to State 0. Count the number of 0s and 1s to verify they meet the criteria after the state machine validation.

Implementation in Code

The state machine approach can be easily implemented in any language. Here's a simple C program to solve the problem:

#include iostream#include cassertlong factorial(int n) {    long product  1L;    while (n  0) {        product * n--;    }    return product;}long choose(int n, int m) {    return factorial(n) / (factorial(m) * factorial(n - m));}void main() {    assert(choose(9, 4)  126);    std::cout  "The number of valid bit strings is: "  choose(9, 3)  std::endl;}

This simple program uses a factorial and a choose function to calculate the number of valid bit strings, which in this case is 84.

Conclusion

Counting specific bit strings with defined patterns is a fundamental problem in combinatorics and algorithm design. Techniques such as block formation, combinatorial methods, and state machines provide powerful tools to solve such problems. The solution to our specific problem not only demonstrates the elegance of combinatorial mathematics but also highlights the importance of algorithmic efficiency in modern computing.