I got 75582.
Explanation:
First, group the 40 identical candies into 20 pairs. It doesn't matter how since the candies are identical. This grouping will ensure that any assigment will contain at least two candies.
Then think of the 20 groups a 20 beads on a string. We are looking to place 11 separators between them to obtain 12 segments, each with a varying number of beads between them. How many ways are there to place 11 separators to 19 potential spaces between beads? The asnwer is
<em>Coplanar</em> means "lying on the same plane," while <em>colinear </em>means "lying on the same line." In this problem, all four points A, B, C, and D are coplanar - they all lie on the plane - so we just need to find the points that lie on the same line. Though points C and B <em>would </em>be colinear if there was a line drawn between them, there's no such line in this problem, so we can rule that out. We notice one line drawn on the plane, and points A, C, and D lying on it, so we can say that points A, C, and D are both coplanar and colinear.
