The exact number of chords that can be drawn through 11 distinct points on a circle is 55.
A chord is defined as a straight line segment that connects any two points on the circumference of a circle. To determine the total number of possible chords, we need to count every unique pair of points that can be selected from the given 11 points. Since the order of selecting the points does not matter (e.g., connecting point A to point B creates the same chord as connecting point B to point A), this is a classic problem of combinations.
Understanding Combinations
The mathematical concept used here is a combination, specifically "n choose k," denoted as C(n, k) or $\binom{n}{k}$. This formula calculates the number of ways to choose k items from a set of n items without regard to the order of selection.
The formula for combinations is:
$C(n, k) = \frac{n!}{k!(n-k)!}$
Where:
- $n$ = Total number of points available
- $k$ = Number of points required to form one chord (always 2)
- $!$ = Factorial (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$)
Calculation for 11 Points
For 11 points on a circle, we have $n = 11$ and $k = 2$.
| Variable | Value | Description |
|---|---|---|
| n | 11 | Total points on the circle |
| k | 2 | Points needed per chord |
Let's plug these values into the combination formula:
$C(11, 2) = \frac{11!}{2!(11-2)!}$
$C(11, 2) = \frac{11!}{2!9!}$
$C(11, 2) = \frac{11 \times 10 \times 9!}{ (2 \times 1) \times 9!}$
$C(11, 2) = \frac{11 \times 10}{2}$
$C(11, 2) = \frac{110}{2}$
$C(11, 2) = 55$
Therefore, a total of 55 unique chords can be drawn through 11 points on a circle.
Distinguishing Chord Formation Scenarios
It's important to clarify the nature of drawing chords. Each chord, by definition, uses precisely two points. When considering the total number of chords that can be drawn, we are looking at every possible unique pairing of the given points.
However, sometimes a different scenario might be considered, such as how many chords can be drawn if each point can only be used once, forming non-overlapping line segments. In such a specific scenario with 11 points, one could form 5 distinct chords (using 10 points as 5 pairs), which would leave one point unconnected. This represents a particular arrangement of chords designed to use as many points as possible without sharing, not an exhaustive count of all possible chords inherent in the set of points. The standard interpretation of the question seeks the latter—all unique pairs.
Further Insights and Examples
- Understanding Point Usage: Every chord utilizes two points. If you have N points, each point can be an endpoint for (N-1) chords. However, summing (N-1) for each point double-counts each chord because each chord has two endpoints, so dividing by 2 leads back to the combination formula.
- Practical Application: This concept is fundamental in geometry and combinatorics. It applies to various scenarios beyond just circles, such as determining the number of line segments that can connect a set of non-collinear points.
For instance, consider a smaller set of points to visualize the calculation:
- 3 points: C(3, 2) = 3 chords. These three chords form a triangle.
- 4 points: C(4, 2) = 6 chords. These form a quadrilateral with its two diagonals.
- 5 points: C(5, 2) = 10 chords. These form a pentagon with all its diagonals.
These examples illustrate that the number of chords grows rapidly with the number of points, highlighting the combinatorial nature of the problem.
Learn More About Combinations
To delve deeper into the mathematical principles behind combinations and permutations, explore reputable resources like Khan Academy's Combinations topic or Wolfram MathWorld's Combinations entry.
