# New results on Identical Binomial Coefficient?

Are there any nontrivial identical binomial coefficients found other than: $${16 \choose 2}={10 \choose 3}=120 \ {21 \choose 2}={10 \choose 4}=210 \ {56 \choose 2}={22 \choose 3}=1540 \ {120 \choose 2}={36 \choose 3}=7140 \ {153 \choose 2}={19 \choose 5}=11628 \ {221 \choose 2}={17 \choose 8}=24310 \ {78 \choose 2}={15 \choose 5}={14 \choose 6}=3003$$ and the infinite family (where $$F_n$$ is the $$n$$th fibonacci number): $${F_{2i+2}F_{2i+3} \choose F_{2i}F_{2i+3}}={F_{2i+2}F_{2i+3}-1 \choose F_{2i}F_{2i+3}+1}$$ By non-trivial I mean, of the shape $${n \choose k}$$ where $$2\leq k \leq \frac{1}{2}n$$. This was conjectured to be the complete list in a paper from 1996: https://pdfs.semanticscholar.org/69fe/a6ef6f5ac5818538a86e98ddd9236a3310f7.pdf Are there any new results?