Corresponding Binary Strings of Odd and Even 1's?

[future_phd]

Homework Statement

Find a one-to-one correspondence between the binary strings (i.e. sequences of 0's and 1's) of length k that have an odd number of 1's, and those that have an even number of 1's.

Homework Equations

The Attempt at a Solution

I'm not exactly sure of what it's asking me to do.
If it's of length k, then you have 2^k different combinations. Half of these will have an even number of 1's, half will have an odd number. So you get (2^k)/2 or 2^(k-1) with an odd number of 1's, and 2^(k-1) with an even number of 1's. Not exactly sure where to go from here.

 

[Mark44]

If I understand this problem correctly, it's very easy. There's no guarantee that I understand it correctly, though.

Consider a binary string of length k, with n 1's and (k - n) 0's. Form a new string of the same length, switching 1's to 0's and 0's to 1's. The new string will have n 0's and (k - n) 1's.

This is clearly a 1:1 mapping, and I think you can satisfy yourself that if the first string had an odd number of 1's, the new string will have an even number of 1's.

 

[future_phd]

If I understand this problem correctly, it's very easy. There's no guarantee that I understand it correctly, though.

Consider a binary string of length k, with n 1's and (k - n) 0's. Form a new string of the same length, switching 1's to 0's and 0's to 1's. The new string will have n 0's and (k - n) 1's.

This is clearly a 1:1 mapping, and I think you can satisfy yourself that if the first string had an odd number of 1's, the new string will have an even number of 1's.

Ahh! That makes sense, that's probably what it's asking. Thanks! :D