Problem Solving

In the rectangular coordinate plane, a quadrilateral region has vertices (2.7,4.2),(-7.2, -1.7),(2.7,-1.7), and (-7.2,4.2). If a point from within the quadrilateral is to be randomly selected such that both the x- and y-coordinates are integers, what is the probability that both coordinates of the selected point are negative?

A. 1/11
B. 8/30
C. 2/15
D. 16/77
E. 14/120

sourabh33 wrote:In the rectangular coordinate plane, a quadrilateral region has vertices (2.7,4.2),(-7.2, -1.7),(2.7,-1.7), and (-7.2,4.2). If a point from within the quadrilateral is to be randomly selected such that both the x- and y-coordinates are integers, what is the probability that both coordinates of the selected point are negative?

A. 1/11
B. 8/30
C. 2/15
D. 16/77
E. 14/120

both x and y coordinates will be negative in the 3rd quadrant..!! also, we are here looking for the integral values of x and y,, therefore integral values of x that will lie in the third quadrant will be (-1,-2,-3,-4,-5,-6,-7) and integral value of y that will lie in the third quadrant will be (-1);

also total no. of integral values of x coordinates bounded by the quadrilateral is (-7,-6,-5,-4,-3,-2,-1,0,1,2) and total integral values of y coordinate is (4,3,2,1,0,-1);

out of 10 integral values of x any 1 can be selected in 10C1 ways;
out of 6 integral values of y any 1 can be selected in 6C1 ways;

therefore total no. of selecting x and y coordinates without any restriction= 10C1*6C1=60;

Required no. of ways (i.e. when x and y coordinates are negative)=7C1*1C1(as there are 7 negative values possible for x-coordinate and only 1 value is possible for y coordinate) hence required probability is 7C1*1C1/10C1*6C1= 7/60 which is option E