Wish for Well

Algebra Level 5

A wishing well is located at the point (11, 11) in the xy-plane. Sanket randomly selects an integer y from the set {0, 1, . . . , 10}. Then he randomly selects, with replacement, two integers a, b from the set {1, 2, . . . , 10}. The probability the line through (0, y) and (a, b) passes through the well can be expressed as m/ n , where m and n are relatively prime positive integers. Compute m + n.


The answer is 111.

Arko Roychoudhury by arko roychoudhury , 20, India

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1 solution

Bill Bell
Oct 8, 2015

Computer scripts have a way of making situations very clear. 

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from itertools import product
from sympy import Point,Line
from fractions import Fraction

wishingWell=Point(11,11)

m=0
n=0
for a,b,y in product(range(1,11),range(1,11),range(0,11)):
    line=Line(Point(a,b),Point(0,y))
    m+=1 if line.distance(wishingWell)==0 else 0
    n+=1
print m,n,Fraction(m,n)

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