Counting active binary combinations

Hi,

I am trying to solve a problem to no avail. I have a 64-bit binary string. The problem requires that only 8-to-16 of these bits can be active (equal to 1). How would I determine the number of such combinations for which this can be true?

I thought I had the solution when I formed the equation:

x=((2^n)-1)*(m-n+1);

Where:

x = number of unique combinations (?),
m = bit string length (64 here), and
n = length of active range (16-8 = 8 here).

However, this only gave the answer for when the active bits were within an 8-bit range of each other in the bit string length. The actual answer must consider situations where those 8-to-16 bits can be anywhere in the 64-bit string.

Please help!

Answer 1

[James Tursa]

You are choosing a combination of 64 bits taken 8-to-16 at a time. So add them up:

s = 0;
n = 64;
for k=8:16
    s = s + nchoosek(n,k);
end

A big number

7.1325e+14

[Walter Roberson]

Right. Or is the question about verifying that a particular given combination only has 8 to 16 bits set??

[Enrico8]

Thank you! That makes perfect sense. I just could not get my head around it. Thanks very much.