# Happy birthday to me (part 2)

Discrete Mathematics Level pending

Let $$S$$ denote the number of ordered $$({n_1}, {n_2},...,{n_{18}})$$ $$18$$-ples solutions to an inequation such that:

$$S=\# \{{n_1},{n_2},...,{n_{18}} \in \mathbb{N_{0}} \mid 1997 < {n_1}+{n_2}+...+{n_{18}} < 2015\}$$.

If the number of solutions can be expressed as $$S=\displaystyle {\binom {a}{b}}-\binom {b-1}{c}$$, find $$\lfloor {x} \rfloor$$, where $$x$$ is the arithmetic mean of $$a,b$$ and $$c$$.

Details and assumptions

• For any $$k\leq{n}$$ there are two different binomial coefficients which satisfy

$$\displaystyle {\binom {n}{k}}=p=\binom {n}{n-k}$$.

In this problem $$c$$ is the smallest possible value;

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