Introduction to the product rule

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SUMMARY

The discussion focuses on the application of the Product Rule in combinatorics to determine the number of bit strings of length 8. It establishes that there are 256 total bit strings, calculated as 2^8. Additionally, it confirms that there are 64 bit strings of length 8 that begin with two 1's, derived from the calculation 2^6 for the remaining six bits. The results are validated by the community, affirming the correctness of the calculations.

PREREQUISITES
  • Understanding of combinatorial principles, specifically the Product Rule.
  • Basic knowledge of binary systems and bit strings.
  • Familiarity with exponentiation and its application in counting problems.
  • Ability to interpret and manipulate mathematical expressions.
NEXT STEPS
  • Study the Product Rule in combinatorics for broader applications.
  • Explore variations of bit strings with different constraints, such as length and starting bits.
  • Learn about the principles of binary counting and its implications in computer science.
  • Investigate other combinatorial techniques, such as the Sum Rule and permutations.
USEFUL FOR

Students in mathematics or computer science, educators teaching combinatorial concepts, and anyone interested in understanding binary string calculations and their applications.

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Homework Statement



a) How many bit strings are there of length 8?

b) How many bit strings are there of length 8 which begins with 2 1's?

Homework Equations



Product Rule

The Attempt at a Solution



a) Since a bit string is either 0 or 1 there are two possibilities for each one. By the rule of products 2^8 = 256 bit strings.

c) the first two choices are fixed so its 11(0/1)(0/1)(0/1)(0/1)(0/1)(0/1)

2^6 = 64 possible different bit strings.

when u allow the preceding 1 to change u basically double the 64 combinations and allowing the first one u quadruple it getting back to the original 256.


Could someone please confirm my result. Danke!
 
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