Algebra... no it's Combinatorics
Consider all pairs ( a , b ) such that a 2 + b 2 ≤ 4 . One of these pairs is randomly chosen, and the probability that a + b ≤ 1 is x . Find the closest integer to 1 0 0 0 x .
The answer is 720.
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1 solution
wohowoo !! By Doing this Question My level is upgraded to Level-5 :)
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Guys. I solved this question by taking the help of calculator. But there is a thing in my mind that can we solve this question without taking help of calculator ,wolfram alpha etc. ???? I don't think it's possible without calculator. @DEEPANSHU GUPTA @Ronak Agarwal
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Of course you can't find inverse tangent functions without a calculator.
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@Ronak Agarwal – Can you please show the method of finding the area bounded by the curve? I'm totally lost :/.
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@Nathan Singiri – 2 ∫ − 2 2 2 4 − x 2 d x
Where y = 4 − x 2 is the equation of a semi circle with radius 2
There is another way of finding the area without calculus but it is very tedious.
I surprised this is level 5 and that this is in combinatorics. I thought it would be in like calculus or geometry.
Oh yes this is simple case of geometrical probability.
All the real pairs satisfying a 2 + b 2 ≤ 4 are points lying inside the circle x 2 + y 2 = 4 in cartesian plane.
We will find the larger area bounded by the curves x 2 + y 2 = 4 and x + y = 1 to find the probability of finding a pair ( a , b ) such that a + b ≤ 1
I won't show the method of finding the area but the expression comes out to be :
A = 2 ( π + t a n − 1 ( 3 7 ) + 4 7 ) ≈ 9 . 0 5 1 5
Hence the probability = T o t a l A r e a A = 4 π 9 . 0 5 1 5 ≈ 0 . 7 2 0 2 9