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

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In summary, an ordered pair (x,y) represents a combination of two numbers chosen from a set of possible options. The equation x choose y = 2020 relates to the ordered pair by representing the number of ways to choose y items from a set of x items. The significance of the number 2020 in this problem is that it represents the total number of possible combinations. This problem can be solved mathematically using formulas and equations, and has real-life applications in areas such as probability, statistics, and computer science.
  • #1
juantheron
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Finding all natural number ordered pair $(x,y)$ for which $\displaystyle \binom{x}{y} = 2020.$
 
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  • #2
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 $(x,y)=(2020,1),(2020,2019)$.
 
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  • #3
Thanks https://mathhelpboards.com/members/olinguito/ My solution is almost same as you.
 

FAQ: Ordered pair (x,y): x choose y = 2020

1. What is an ordered pair?

An ordered pair is a pair of numbers written in a specific order, usually in the form (x,y). The first number, x, is known as the x-coordinate and the second number, y, is known as the y-coordinate.

2. What does "x choose y" mean?

"x choose y" is a mathematical notation that represents the number of ways to choose y items from a set of x items. It is also known as a binomial coefficient and is often written as xCy.

3. How do you calculate "x choose y"?

The formula for calculating "x choose y" is x! / (y! * (x-y)!), where ! represents the factorial function. This can also be written as xCy = x! / (y! * (x-y)!).

4. What is the significance of "x choose y = 2020"?

This equation is often used in combinatorics, which is the branch of mathematics that deals with counting and arranging objects. In this case, it represents the number of ways to choose y objects from a set of x objects, where the result is equal to 2020. It could also be used in problem-solving or probability questions.

5. Can "x choose y" be used for any values of x and y?

Yes, "x choose y" can be used for any positive integers x and y, as long as x is greater than or equal to y. The result will always be a non-negative integer.

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