You can draw exactly 210 chords through 21 points on a circle.
Understanding Chords and Combinations
When you have a set of distinct points arranged on the circumference of a circle, a chord is simply a straight line segment that connects any two of these points. To determine the total number of unique chords that can be drawn, we need to identify how many different pairs of points can be selected from the given set.
This type of problem is a fundamental concept in combinatorics, a branch of mathematics focused on counting, arrangement, and combination. Since the order in which you select two points does not change the resulting chord (e.g., a chord from point A to point B is identical to a chord from point B to point A), we use the combination formula.
The Combination Formula
The formula for calculating the number of combinations, often read as "n choose k," is denoted as nCk or C(n, k):
nCk = n! / (k! * (n-k)!)
Where:
nrepresents the total number of distinct items available (in this case, the total number of points on the circle).krepresents the number of items to choose for each group or selection (for a chord, we always choose 2 points).!signifies the factorial operation, which means multiplying a number by all the positive integers less than it (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120).
Calculating Chords for 21 Points
To find the number of chords that can be drawn through 21 points on a circle, we apply the combination formula with our specific values:
- Total number of points (
n) = 21 - Number of points needed to form a single chord (
k) = 2
Let's plug these values into the formula:
21C2 = 21! / (2! * (21-2)!)
21C2 = 21! / (2! * 19!)
Now, we expand the factorials:
21C2 = (21 × 20 × 19 × 18 × ... × 1) / ((2 × 1) × (19 × 18 × ... × 1))
Notice that 19! appears in both the numerator and the denominator, allowing us to cancel it out:
21C2 = (21 × 20) / (2 × 1)
21C2 = 420 / 2
21C2 = 210
Thus, there are 210 distinct chords that can be drawn through 21 points on a circle.
Calculation Summary
| Parameter | Value |
|---|---|
Total Points (n) |
21 |
Points per Chord (k) |
2 |
| Combination Formula | 21C2 |
| Total Number of Chords | 210 |
Practical Insights and Examples
The concept of combinations is incredibly versatile and extends far beyond geometry. It's fundamental in areas like probability, statistics, and computer science for calculating possible arrangements or selections.
-
Smaller Scale Example: Consider a circle with just 4 points (let's call them A, B, C, D).
- Using the formula:
4C2 = 4! / (2! * (4-2)!) = 4! / (2! * 2!) = (4 × 3 × 2 × 1) / ((2 × 1) × (2 × 1)) = 24 / 4 = 6chords. - You can manually list them: AB, AC, AD, BC, BD, CD – confirming 6 chords.
- Using the formula:
-
Understanding the Logic: Each of the 21 points can connect to 20 other points. If you multiply 21 by 20, you get 420. However, this counts each chord twice (e.g., AB and BA are counted separately). Therefore, dividing by 2 corrects this double-counting, leading to 420 / 2 = 210.
To learn more about combinations and permutations, you can explore resources like Khan Academy's explanation of combinations.
