Question Number 212066 by Spillover last updated on 28/Sep/24
Answered by Rasheed.Sindhi last updated on 28/Sep/24
$$\frac{\mathrm{2021}!+\mathrm{2020}!}{\mathrm{2021}!−\mathrm{2020}!}\:\centerdot\:\frac{\mathrm{2020}!+\mathrm{2019}!}{\mathrm{2020}!−\mathrm{2019}!}\:\centerdot…\frac{\mathrm{3}!+\mathrm{2}!}{\mathrm{3}!−\mathrm{2}!}\:\centerdot\:\frac{\mathrm{2}!+\mathrm{1}!}{\mathrm{2}!−\mathrm{1}!} \\ $$$$\frac{\left\{\left({x}+\mathrm{1}\right)!+{x}!\right\}}{\left\{\left({x}+\mathrm{1}\right)!−{x}!\right\}}=\frac{{x}!\left\{\left({x}+\mathrm{1}\right)+\mathrm{1}\right\}}{{x}!\left\{\left({x}+\mathrm{1}\right)−\mathrm{1}\right\}}=\frac{{x}+\mathrm{2}}{{x}} \\ $$$$\frac{\mathrm{2020}+\mathrm{2}}{\mathrm{2020}}\centerdot\frac{\mathrm{2019}+\mathrm{2}}{\mathrm{2019}}\centerdot…\frac{\mathrm{2}+\mathrm{2}}{\mathrm{2}}\centerdot\frac{\mathrm{1}+\mathrm{2}}{\mathrm{1}} \\ $$$$\frac{\left(\mathrm{2020}+\mathrm{2}\right)\left(\mathrm{2019}+\mathrm{2}\right)\left(\mathrm{2018}+\mathrm{2}\right)…\left(\mathrm{2}+\mathrm{2}\right)\left(\mathrm{1}+\mathrm{2}\right)}{\mathrm{2020}.\mathrm{2019}.\mathrm{2018}….\mathrm{2}.\mathrm{1}} \\ $$$$=\frac{\mathrm{2022}.\mathrm{2021}.\mathrm{2020}….\mathrm{4}.\mathrm{3}}{\mathrm{2020}!} \\ $$$$=\frac{\mathrm{2022}.\mathrm{2021}.\mathrm{2020}….\mathrm{4}.\mathrm{3}.\left(\mathrm{2}.\mathrm{1}\right)}{\mathrm{2020}!\left(\mathrm{2}.\mathrm{1}\right)} \\ $$$$=\frac{\mathrm{2022}.\mathrm{2021}}{\mathrm{2}}=\mathrm{2043231} \\ $$