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- Thread starter jacks
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- Apr 22, 2018

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My attempt (may be wrong).

Given a binomial coefficient $\displaystyle\binom xy$, we can make two observations:

But $\displaystyle\binom{101}2\ =\ 5050\ >\ 2020$.

It follows from the two observations above (and the fact that $\displaystyle\binom xy=\binom x{x-y}$) that $\displaystyle\binom xy>2020$ for all $x\ge101$ and $y=2,3,\ldots,x-2$.

Hence the only integers $x,y$ such that $\displaystyle\binom xy=2020$ are $(x,y)=(2020,1),(2020,2019)$.

- For fixed $x$, $\displaystyle\binom xy$ increases as $y$ increases from $0$ to $y=\left\lfloor\dfrac x2\right\rfloor$.
- For fixed $y=1,\ldots,\left\lfloor\dfrac x2\right\rfloor$, $\displaystyle\binom xy$ increases as $x$ increases.

But $\displaystyle\binom{101}2\ =\ 5050\ >\ 2020$.

It follows from the two observations above (and the fact that $\displaystyle\binom xy=\binom x{x-y}$) that $\displaystyle\binom xy>2020$ for all $x\ge101$ and $y=2,3,\ldots,x-2$.

Hence the only integers $x,y$ such that $\displaystyle\binom xy=2020$ are $(x,y)=(2020,1),(2020,2019)$.

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