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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