Discussion Overview
The discussion revolves around the combinatorial problem of determining how many 8-bit bytes contain exactly two 1's. Participants explore different approaches to calculating this quantity, including the use of binomial coefficients and permutations.
Discussion Character
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant questions the answer key's use of \binom{8}{2}, suggesting that after selecting two bits for 1's, there are still 2^6 ways to arrange the remaining bits, which seems larger than 28.
- Another participant explains that there are 8 positions for the first 1 and 7 for the second, but since order does not matter, the calculation should be divided by 2, leading to the conclusion that there are 28 valid combinations.
- A similar explanation is reiterated by another participant, emphasizing the reasoning behind dividing by 2 to avoid double counting permutations of the same selection.
- One participant notes that once the two bits for 1's are chosen, the remaining bits must all be 0's, indicating that there is only one way to arrange the remaining bits.
- A later reply expresses a personal feeling of confusion and fatigue regarding the problem, indicating a struggle with understanding the reasoning involved.
Areas of Agreement / Disagreement
Participants express differing views on the initial reasoning behind the problem. While some agree with the combinatorial approach leading to 28, others remain uncertain about the validity of their initial interpretations.
Contextual Notes
There is a lack of consensus on the initial misunderstanding of the problem, with some participants not fully grasping why their approach differs from the combinatorial solution provided.