Floor function and arithmetic sequences in combinations

Algebra Level 5

\[\large S = \binom{3}{\lfloor 1 \rfloor}+\binom{6}{\lfloor 2.5 \rfloor}+\binom{9}{\lfloor 4 \rfloor}+\cdots + { 3n \choose \lfloor 1.5n - 0.5 \rfloor } + \cdots +\binom{99}{\lfloor 49 \rfloor}\]

Find the remainder when \(S\) is divided by 1000.

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