S is the set of points each of whose whose coordinates x, y and z is an integer that satisfy:
0 ≤ x ≤ 2, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 4

Two distinct points are randomly chosen from S.

Determine the probability that the midpoint of the segment that they determine also belongs to S.

The x, y, and z values of the two points have to have the same parity. There are 5 possibilities for the x values, 8 for the y values, and 13 for the z values. Therefore, there are 5*8*13=520 pairs of points with the same parity. There are 3*4*5=60 pairs that are the same point, so there are 520-60=460 that are distinct. There are 60*59=3540 ways of picking two distinct points. Therefore, the probability is 460/3540=23/177.

Posted by Math Man on 2014-08-30 21:11:41