I was going through a JSTSE paper, When I found this question...

How many points (x,y) exist in the 1st quadrant such that x+y=7 ? Note by Mehul Arora
6 years, 4 months ago

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If the number of points(x,y) are said to be only whole numbers(not decimals), then there are 6, they are:

$(1,6),(2,5),(3,4),(4,3),(5,2) and (6,2)$

If it is mentioned, including decimals, then it is infinite . . .

What do you say Azhaghu Roopesh M ???

- 6 years, 4 months ago

Yeah , since the number 7 is quite small , we can count it out .

- 6 years, 4 months ago

Exactly. Is there any formula for these kind of problems???

- 6 years, 4 months ago

Look up the Generating Functions wiki here on B'ant . Anyway , I've got to go now, Bye :)

- 6 years, 4 months ago

@Sravanth Chebrolu Do (0,7) and (7,0) not lie in any quadrant? Coz the definition of the point in the 1st quadrant is x,y>0. Anyways.. If included, they should be 8 in number , eh. I think there is a typo.

- 6 years, 4 months ago

Yeah it was a typo, I've changed it. And yes, I think what you said is also right, the points (7,0) and (0,7) lie on the axes not the quadrants.

- 6 years, 4 months ago

The formula for the number of Natural numbers satisfying $x + y = k$ is always $k-1$.
So in this case k = 7 . Hence number of points are 6

This formula is just a observation based. Please verify it @Azhaghu Roopesh M @Krishna Ar @Sravanth Chebrolu

- 6 years, 4 months ago

Yes, I too think the same, but if the no. of points asked is in fractions or decimals there can be many . . .

What do you say @Rajdeep Dhingra ???

- 6 years, 4 months ago

Well as this formula is valid only for natural numbers so the sum will also be a natural number.
If it is not specified natural number or k is a rational number with denominator $\ne$ 1 then the answer is $\infty$

- 6 years, 4 months ago

Exactly. Then this might have been included in the question . . .

- 6 years, 4 months ago

Is this the exact same question ? If so , it is a simple question of Generating Functions

- 6 years, 4 months ago

Generating functions? Are you sure? This is a JSTSE question, so its for classes 10 or below I think.

- 6 years, 4 months ago

Not exactly below.. The maths questions were mainly Of class 10 level iincluding all AP, GP , Quadratics, Vieta's etc...

- 6 years, 4 months ago

Well , I think any class 10 student has the ability to count the number of solutions manually , it's just that the method of Generating Functions works for finding solutions of the type :$x+y = N$ or other types .

- 6 years, 4 months ago

I guess there would have been more information provided. Most likely the question would have specified $x,y$ to be integers.

- 6 years, 4 months ago

But it has been said that x,y belong to the first quadrant , so automatically $x,y \geq 0$

- 6 years, 4 months ago