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Counting Combinations

Many combinatorial and algebraic questions arise in the investigation of geometric figures; these can be introduced at different educational levels, right up to the frontiers of research. How many edges does a triangular pyramid have? We can follow Froebel's suggestion and make a model out of toothpicks and peas, then count the edges. Or we can simply draw a picture of the object (Figure 40) and count the six edges.

The procedure for drawing such a diagram suggests an algorithm for determining the number of edges. Start with a point, then choose a distinct point and draw the one edge connecting it to the one we already had. Now choose a new point and connect it to the previous two points to get two more, for a total of three. (We have to be careful not to choose the new point on the line containing a previous edge.) Next choose a new point not lying on any of the three lines determined by the edges already constructed, and then connect this new point to the previous three. This yields three new edges, for a total of six.

Figure 40. The tetrahedron — the simplest regular polyhedra — has four triangular faces, six edges, and four vertices.

Figure 41. By adding one new point with line-segment connections to each previous vertex, one can construct in sequence the complete graphs on 1, 2, 3, 4, 5, 6, ... points.

We can repeat this process to draw the figure-called a complete graph — determined by five points (Figure 41). First choose a point not on any of the six lines containing previously constructed edges, and then connect it to the previous four points to obtain four new edges — for a total of 10. A similar construction can produce the complete graph on six points and more if so desired.

What is the pattern that emerges from this procedure? It becomes apparent if we arrange the results in a table:

Number of points:123456
Number of edges:01361015

In each case the number of edges is the number of pairs of points, which leads directly to the study of combinations. Based on the sequence of construction, it is easy to see that the number of edges at stage n is the sum of all numbers less than n. For example, the number of edges formed by six points is 1 + 2 + 3 + 4 + 5 = 15. Some students may know the formula n(n + 1) / 2 for the sum of the first n integers, perhaps in conjunction with the famous story of the young Gauss who used this formula to add up all the numbers from 1 to 100. Another type of pattern is revealed by the table — that the number of edges at any stage is the total of the previous number of edges and the previous number of vertices. [an error occurred while processing this directive]