Does Stars and Bars or the binomial coefficient represent binary sequences?
With the binomial coefficient we can calculate all the paths on a grid with moving up or right, that's like defining up to be $1$ and right to be $0$, so the binomial coefficient gives the amount of binary sequences of size $nk$ (?)
With Stars and Bars, by defining the stars to be zeros and bars to be ones, does it tell anything about a binary sequence?
On
The binomial coefficient gives the number of lattice paths to a fixed point, not those of a fixed length. There are $2^n$ lattice paths or binary sequences of length $n$; there's no need for either binomial coefficients or stars and bars.
On
The coefficient ${n \choose k}$ counts the number of binary sequences of length $n$ with exactly $k$ $0$'s (or, exactly $k$ $1$'s, if you prefer).
I can't see stars and bars being inherently useful for binary sequences, but I could be wrong. They're binomial coefficients as well, but why the number of (non-negative integer) solutions to $x_1 + x_2 + x_3 = 5$ should count the number of sequences of length $7$ with exactly $2$ zeros seems less obvious to me.
If a binary sequence is a sequence of digits selected from $\{0,1\}$, then the binomial coefficient $\binom nk$ is the number of binary sequences of exactly $n$ digits that contain exactly $k$ zeros.
"Stars and bars" is generally applied when we want to put a number of indistinguishable items into a number of identified bins, or in problems equivalent to that. The digits of a binary sequence are distinguishable, but of only two kinds, so it's hard to see a good application of stars and bars there.
Of course "stars and bars" itself evaluates to a binomial coefficient, which one could apply to binary sequence as already shown, but it's not clear why one would call this "stars and bars".