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At least 3 zeroes (Posted on 2018-10-03) Difficulty: 3 of 5
How many binary strings of length n, having three or more zeroes, exist?

No Solution Yet Submitted by Ady TZIDON    
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solution | Comment 3 of 5 |

A qualifying string contains somewhere between m = 0 and n-3 “1”s. These m 1s may be placed in any of n positions. So total number of strings T becomes:

T = sum( {m=0 -> n-3}  (n)! / [(n-m)! m!] )  

We are summing over binary coefficients (n over m) usually called (n choose m)

I have verified this result using strings, and it also agrees with C and D’s solutions. While neither as direct nor as easy a route as theirs, it works (and may be readily generalized to any number of zeros)! 

Edited on October 3, 2018, 6:03 pm
  Posted by Steven Lord on 2018-10-03 12:36:52

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