MHB How many bytes contain exactly two 1's?

  • Thread starter Thread starter find_the_fun
  • Start date Start date
  • Tags Tags
    bytes
AI Thread Summary
The discussion centers on calculating the number of byte configurations containing exactly two 1's. The correct approach involves using the combination formula \binom{8}{2}, which accounts for selecting 2 positions from 8 without regard to order, resulting in 28 unique combinations. The confusion arises from misunderstanding that once the two 1's are placed, the remaining 6 bits must be 0's, which only allows for one arrangement of those bits. The explanation clarifies that while there are multiple ways to arrange bits, the specific requirement of having exactly two 1's limits the configurations. Ultimately, the key takeaway is that the total number of valid byte configurations is indeed 28.
find_the_fun
Messages
147
Reaction score
0
How many bytes contain exactly two 1's?

The answer key give [math]\binom{8}{2}[/math] and I don't get why. Even if two bytes have been taken there are still [math]2^6[/math] ways to arrange the other bytes, which is much larger than 28.
 
Physics news on Phys.org
You could look at it this way:

There are 8 bit positions you could choose for the first 1 and that leaves 7 for the second bit position. But, order does not matter, so you want to divide by 2 so that you do not count each possible permutation twice. For example, suppose you chose the third bit the first time and the fifth bit the second time. This is equivalent to choosing the fifth bit the first time and the third bit the second time. So, we find:

$$N=\frac{8\cdot7}{2}=28$$
 
MarkFL said:
You could look at it this way:

There are 8 bit positions you could choose for the first 1 and that leaves 7 for the second bit position. But, order does not matter, so you want to divide by 2 so that you do not count each possible permutation twice. For example, suppose you chose the third bit the first time and the fifth bit the second time. This is equivalent to choosing the fifth bit the first time and the third bit the second time. So, we find:

$$N=\frac{8\cdot7}{2}=28$$

What I don't see is why the way I was looking at it was wrong.
 
If you have already chosen the two bits that are 1, then the remaining 6 bits must be zeroes, and there is only 1 way this can be.
 
MarkFL said:
If you have already chosen the two bits that are 1, then the remaining 6 bits must be zeroes, and there is only 1 way this can be.

That's what I was waiting for someone to say! I don't know what's wrong with me I need more sleep.
 
Back
Top