I have a problem where I need to find the total number of bit-strings with the length of 30 that start with 3 ones and end with 2 ones. The total number of 1s in the string is 17 and the total number of 0s is 13. It would look something like this, with 25 empty spots in the middle of the bit-string:
1 1 1 _ ... _ 1 1
My initial guess for solving this is to find all possible combinations for the remaining empty spots in the bit strings with: $${25 \choose 12\quad 13} $$ I am not sure if this solution is correct or if I should apply the inclusion-exclusion principle. If I need to apply that principle, how would it be done?

  • $\begingroup$ I think your string should look like $111\underbrace {\cdots\cdots }_{25\; \text {terms}}00$, no? In which case there are $25$ empty slots and you just have to pick the places to put the $11$ non-tail $0's$ so... $\endgroup$ – lulu Sep 15 '18 at 22:28
  • $\begingroup$ @lulu my mistake, I actually had the string written correctly but made the mistake in the title. Edited it. $\endgroup$ – D. D. Sep 15 '18 at 22:33
  • $\begingroup$ Your answer is correct, although it is normally written in one of the following forms $\binom{25}{12} = \binom{25}{13} = \binom{25}{12, 13}$. $\endgroup$ – N. F. Taussig Sep 15 '18 at 22:35
  • $\begingroup$ @N.F.Taussig Thank you! $\endgroup$ – D. D. Sep 15 '18 at 22:40
  • $\begingroup$ Never mind, I was answering the older form of the question. Post edit, of course the present form is correct. $\endgroup$ – lulu Sep 15 '18 at 22:43

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