How many bit strings of length 10 contain exactly 5 bits equal to 1? Is it 2^10 - 2^5?

Question

turingtest · Answer

I have no idea what this means :(

turingtest · Answer

you have some number of bits of string length 10
and you want to know "how many contain 5 bits equal to 1" ??
what do you mean, 5 pieces of string equal to one what?

anonymous · Answer

Sorry, this was gotten straight from the problem statement. I'll rephrase this a little bit. How many 10-bit strings are there that have exactly five 1's in them?

turingtest · Answer

@asnaseer can you lend a hand?

anonymous · Answer

the 1's can be in any nth bit. doesn't have to be next to each other :D

turingtest · Answer

my impulse is to agree with you, 2^10-2^5, but combinatorics is not my strong suit

anonymous · Answer

Thank you. And it's alright. :D

phi · Answer

isn't it just 10 choose 5 ?
$$\left(\begin{matrix}10 \ 5\end{matrix}\right)=\frac{10!}{5! 5!}$$

anonymous · Answer

Hm. I'm a little bit fuzzy with combinations right now. 10 choose 5 because..?

phi · Answer

does this explain it? http://answers.yahoo.com/question/index?qid=20091122040726AAi6YU0

phi · Answer

Or http://answers.yahoo.com/question/index;_ylt=AhmQsH93Xi0bJtqIFdcUKLMCxgt.;_ylv=3?qid=20080504195059AAir3Ep the intuition is that you have 10 spaces all filled with 0's. You choose 5 of the zeros to be replaced with a 1. How many ways can we choose 5 out of 10 ? 10C5 is the standard answer

anonymous · Answer

haha I think that explanation suffices. thank you very much!