Problem of the Day #34

POTD January 8, 2024
Note that POTD structure was changed again to have a random point difficulty every day

Find the remainder when \binom{4}{2}\binom{2}{2}+\binom{5}{2}\binom{3}{2}+\binom{6}{2}\binom{4}{2}+\binom{7}{2}\binom{5}{2}+ ...\binom{50}{2}\binom{48}{2} is divided by 1000.