Question

Find the number of chords that can be drawn through 16 points on a circle. (a) 102 (b) 120 (c) 12 (d) \({ }^{16} \mathrm{P}_{2}\)

Short answer

The number of chords that can be drawn through 16 points on a circle is 120. The correct answer is (b) 120.

Step-by-step solution

Understand the problem

We have 16 points on a circle, and we have to find the number of chords that can be drawn through these points. To form a chord, we need to choose 2 distinct points on the circle.

Use the combination formula

Since we need to choose 2 points out of 16 and order does not matter, we can use the combination formula (nCr), which is: \[ \mathrm{nCr} = \frac{n!}{r!(n-r)!} \] where n represents the total number of items, r represents the number of items we want to choose, and ! denotes a factorial.

Substitute values in the combination formula

Now, we need to plug in the values into the combination formula. In this case, n=16 and r=2. \[ ^{16}\mathrm{C}_{2} = \frac{16!}{2!(16-2)!} \]

Calculate the factorial values

First, let's calculate the factorial values of 16, 2, and 14. \[ 16! = 16 \times 15 \times 14! \\ 2! = 2 \] Since 14! is common in both the numerator and denominator, we can eliminate it.

Calculate the number of chords

Now, we can find the number of chords. \[ ^{16}\mathrm{C}_{2} = \frac{16!}{2!(16-2)!} = \frac{16 \times 15 \times 14!}{2 \times 14!} = \frac{16 \times 15}{2} = 8 \times 15 = 120 \] So, the number of chords that can be drawn through 16 points on a circle is 120. The correct answer is (b) 120.

Key concepts

Combinations in Mathematics

In mathematics, combinations refer to the selection of items from a collection where the order does not matter. It's a core concept in many probability and statistics problems, where you want to know how many different groups can be formed from a larger set.

For example, if you have a deck of cards and you want to know how many different 5-card hands are possible, you would use combinations to calculate this. The notation for combinations is often written as \( nCr \) or \( C(n, r) \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items you want to select.

Using the formula for combinations, \( nCr = \frac{n!}{r!(n-r)!} \), allows us to find all possible groups of a certain size without worrying about their order. This is particularly useful when dealing with problems like finding the number of chords in a circle, formed by selecting points on the circumference. Each chord is determined by two endpoints, and since the order of points doesn’t matter (a chord from point A to B is the same as from point B to A), combinations are the right tool for the job.

Factorial Notation

The factorial notation is an essential mathematical concept commonly represented by an exclamation mark (!) after a number. It's defined as the product of all positive integers from 1 up to that number. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

How Factorials Simplify Combinations

Factorials play a significant role in simplifying combination problems. In the combination formula, factorials help in figuring out the number of ways to arrange a subset of a larger set, where order doesn’t matter. When calculating combinations, the denominator often has factorial terms that cancel out similar factors in the numerator, making computations more manageable.

As seen in the chord problem, \( 16! \) over \( 14! \) allows us to easily simplify because the \( 14! \) factors cancel each other out, leaving only the remaining terms to be computed. This reduces the computational complexity and helps in arriving at the answer much more quickly.

Permutation and Combination

Permutation and combination are two fundamental ways of counting arrangements in a set and are a part of combinatorial mathematics.

Understanding Permutations

Permutations take into account the arrangement of items, which means the order is significant. For example, the arrangements ABC, ACB, BAC, BCA, CAB, and CBA are all different permutations of the letters A, B, and C.

Distinguishing Combinations

In contrast, combinations consider the selection of items without concern for order. In the context of combinations, the selections ABC, ACB, BAC, BCA, CAB, and CBA would all be regarded as the same combination.

The formulas for permutations (\(nPr\)) and combinations (\(nCr\)) are closely related, but their applications are distinct based on whether the order of selection is important. Understanding when and how to apply these principles is crucial for solving a variety of mathematical problems, from the number of different ways to arrange books on a shelf (permutations) to the number of chord possibilities in a circle (combinations).