# 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.

×