Evaluate $\frac{2021!+2020!}{2019!+2018!}.$

Question

Evaluate $\displaystyle{\frac{2021!+2020!}{2019!+2018!}}.$

I think we can write this as $\displaystyle{\frac{2018!\cdot2019\cdot2020\cdot2021+2018!\cdot2019\cdot2020}{2018!\cdot2019+2018!}},$ but I don't know if this is the right direction.

Answer 1

$$\frac{2021!+2020!}{2019!+2018!}$$

$$ =\frac{2022\cdot2020!}{2020\cdot2018!}$$

$$= \frac{2022\cdot2020\cdot2019}{2020}$$

$$= 2022\cdot2019$$

Answer 2

Evaluate $\displaystyle{\frac{2021!+2020!}{2019!+2018!}}$.

$$=\displaystyle{\frac{2018!\cdot2019\cdot2020\cdot2021+2018!\cdot2019\cdot2020}{2018!\cdot2019+2018!}}=\displaystyle{\frac{2019\cdot2020\cdot2021+2019\cdot2020}{2019+1}}=2019\cdot2020\displaystyle{\left(\frac{2021+1}{2020}\right)}=2019\cdot 2022=4082418$$