Problem 77 - Precalculus with Limits

Question

Three points that are not collinear determine three lines. How many lines are determinedby nine points, no three of which are collinear?

Step-by-Step Solution

Answer

The nine points can determine 36 lines.

Step 1: Identify The Relevant Formula

Since we're trying to find the number of lines determined by nine points, the formula we will need is that for the number of combinations of two objects selected from n objects, which is \( \binom{n}{2} \). This formula comes from the general combination formula, \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where n is the total number of objects, k is the number of objects chosen at a time, and '!' denotes factorial which means the product of all positive integers less than or equal to n.

Step 2: Plug The Parameters Into The Formula

Here, we have 9 points; therefore, \( n = 9 \) and since a line is defined by 2 points, \( k = 2 \). Substituting these values into the formula, we have \( \binom{9}{2} = \frac{9!}{2!(9-2)!} \) .

Step 3: Solve The Factorials

Factorials can have large values, so it is usually easier to simplify first. We have \( 9! = 9*8*7*6*5*4*3*2*1 \) and \( 2! = 2*1 \) which will simplify to \( 7! = 7*6*5*4*3*2*1 \). Now we put these findings into the formula: \( \binom{9}{2} = \frac{9*8*7!}{2*1*7!} \). As we can see, \( 7! \) will cancel out, leaving us with \( \frac{9*8}{2} \).

Step 4: Perform The Final Calculation

Doing the division first (according to the order of operations), we get \( \frac{9*8}{2} = 9*4 = 36 \). So there are 36 lines that can be determined by nine distinct points.