How many bit strings of length 10 can the bits of "1" be in pairs? Ie: 0110000110 has the property but 0011100110 does not)
My solution to this was to group the two 1bits into a block and the maximum pairs can only be 3 pairs in separated positions in a string of length 10. Then I choose the middle spaces between a string of zeros like this:
But The correct answer given was
What I don't understand is why is there ? Why is there a 11 choose 0, which creates an additional 1 more possibility?
oh yea...sorry... typo on that one. You are right, it should be 41 for the correct answer.
If the term counts the string of all zeros, wouldn't it be wrong because the question only wanted the bits to be in pairs of "1"s? A string of all zeros have no pairs of "1"s but why is it being counted into the total ways?
Maybe I am not used to this yet. I kind of still feel weird because then I could do the same to all, like not just but also add , and because they all are false statements that is true. Then this would result to an even larger possible ways.
hmm.. The only funny thing is in a 10-bit string, 0000000000, if all the places are already occupied by zeros, then there is no way I can have an isolated pair of 1s. The only way I can have a pair of 1s is to drop 2 zeroes to free up 2 "seats" for the pair of 1s. Otherwise, I could only imagine that despite all the "seats" are occupied by zero, there is still a pair of 1s in the "basket" that I haven't picked because I couldn't pick it, which this essentially makes that extra 1 more possbility.