Chapter 2.1, Problem 50HP

(a)

To determine

To Find: The conjecture about the relationship between the number of pointson the circle and the number of regions formed in the circle.

Answer to Problem 50HP

The points divide the circle in $2^{n-1}$ regions.

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider that for $n=2$ the number of regions is 2 and for $n=3$ the number of regions is 4.

Thus, for $n=4$ regions the sketch is shown in Figure 2

  

Figure 2

The above diagram shows that for $n=4$ the number of point is 8 and this shows that the n points divide the circle in $2^{n-1}$ regions.

(b)

To determine

To Find: Theway in which the conjecture hold true when there are six points.

Answer to Problem 50HP

The conjecture is that the n points divide the circles in $n{C}_4+n{C}_2+1$ regions.

Explanation of Solution

Consider the number of points is $n=6$ then the figure for it is shown in Figure 4

  

Figure 4

The number of regions is $2^{6-1}=32$ this shows that the conjecture is not correct. Thus the true conjecture is that the n points divide the circles in $n{C}_4+n{C}_2+1$ regions.