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ordered pair (x,y): x choose y = 2020

jacks

Well-known member
Apr 5, 2012
226
Finding all natural number ordered pair $(x,y)$ for which $\displaystyle \binom{x}{y} = 2020.$
 

Olinguito

Well-known member
Apr 22, 2018
251
My attempt (may be wrong).

Given a binomial coefficient $\displaystyle\binom xy$, we can make two observations:
  • 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.
Now $\displaystyle\binom xy=2020$ $\implies$ $x!=2020\cdot y!\cdot(x-y)!$. So the prime $101$ divides $x!$ since it divides $2020$. Thus we must have $x\ge101$.

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 $(x,y)=(2020,1),(2020,2019)$.
 
Last edited:

jacks

Well-known member
Apr 5, 2012
226
Thanks Olinguito My solution is almost same as you.