Counting binary sequences with $S$ $0$'s and $T$ $1$'s where every pre-sequence contains fewer $1$'s than $0$'s How many $S+T$-digit binary sequences with exactly $S$ $0$'s and $T$ $1$'s exist where in every pre-sequence the number of $1$'s is less than the number of $0$'s?
Examples:


*

*the sequence $011100$, is bad since the pre-sequence $011$ has more $1$'s than $0$'s.

*the sequence $010101000$, is good since there is no pre-sequence such that there are more $1$'s than $0$'s.

This is a famous problem often called Bertand's Ballot Theorem. A good summary is given in the Wikipedia article cited. There are a number of nice proofs.
Note that your statement is the classical one ("always ahead") but the example of a good sequence that you give shows that "never behind" is intended. If that is the case, go to the "ties allowed" section of the article. The number of good sequences turns out to be
$$\binom{s+t}{s}\frac{s+1-t}{s+1}.$$
