How many bit-strings can be formed with ones and zeros?*

Question

How many bit-strings can be formed with 6 ones and 28 zeros if each line must start with 1 and after each one must be at least three 0? Sombebody please help!

Answer 1

So assuming this reflects the limits on a valid solution:

$$\color{red}{\fbox{1000}}\;0\;\color{blue}{\fbox{1000}}\;\color{blue}{\fbox{1000}}\;0\;0\;\color{blue}{\fbox{1000}}\;0\;0\;0\;\color{blue}{\fbox{1000}}\;0\;0\;\color{blue}{\fbox{1000}}\;0\;0 $$

where the initial $\color{red}{\fbox{1000}}$ block is fixed at the start (so produces no variations) and the $5$ $\color{blue}{\fbox{1000}}$ blocks and the $10$ spare zeroes are free to shuffle.

Thus you are selecting $5$ of the $15$ "floating" elements to be $\color{blue}{\fbox{1000}}$ blocks: $${15 \choose 5} = 3003 \text{ options}$$