Question

In how many ways can you select two people from a group of 20 if the order of selection is not important?

Short answer

Answer: There are 190 ways to select two people from a group of 20 when the order of selection does not matter.

Step-by-step solution

Understanding Combinations

Combinations refer to selecting items from a larger set such that the order of selection does not matter. In other words, it's a selection where the elements have no order. In this exercise, we need to select two people from a group of 20 people. Since the order of selection is not important, we can use the combinations formula to solve this problem. The notation we use for a combination is 'C(n, r)', where 'n' is the number of items to choose from, and 'r' is the number of items we want to choose.

Formula for Combinations

The formula for combinations is given by C(n, r) = \frac{n!}{r!(n-r)!}, where '!' denotes factorial ('n!' means you multiply all positive integers from 1 up to 'n', i.e., 5! = 5 × 4 × 3 × 2 × 1 = 120).

Applying the Combinations Formula

Now, let's apply the combinations formula to find the number of ways to select two people from a group of 20. We have 'n' = 20 (the total number of people) and 'r' = 2 (the number of people we want to select). So, we can plug these values into the combinations formula: C(20,2) = \frac{20!}{2!(20-2)!}.

Calculate the Factorials

Next, we need to calculate the factorials in the formula: 1. 20! = 20 × 19 × 18 × ... × 3 × 2 × 1 2. 2! = 2 × 1 3. (20-2)! = 18! = 18 × 17 × 16 × ... × 3 × 2 × 1

Simplify and Solve

Now, let's simplify and solve the expression: C(20,2) = \frac{20!}{2!(20-2)!} = \frac{20!}{2!18!} We can cancel out some terms in the expression to simplify it: C(20,2) = \frac{20 × 19 × 18 × ... × 3 × 2 × 1}{(2 × 1)(18 × 17 × 16 × ... × 3 × 2 × 1)} Since 18 and remaining terms are in both the numerator and denominator, we can cancel them out: C(20,2) = \frac{20 × 19}{2 × 1} = \frac{380}{2} = 190 So, there are 190 different ways to select two people from a group of 20 when the order of selection is not important.

Key concepts

Factorials

Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. A factorial of a number, denoted by an exclamation mark (!), is the product of all positive integers up to that number. For example, the factorial of 5 is written as 5! and calculated as 5 × 4 × 3 × 2 × 1 = 120.
Factorials are crucial in combinations because they help calculate the total number of ways to order a set of items, and in the context of combinations, they help determine the different ways to select items without regard to order.
For a given number \( n \), the factorial \( n! \) can be calculated as follows:

• Start with the number \( n \).
• Multiply it by each consecutive integer less than itself, down to 1.

Factorials grow quickly as \( n \) increases, and converting these into calculations using formulas is essential in combinatorics.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and analyzing possibilities and structures. It is often focused on finite sets or countable discrete structures. Among its many applications, combinatorics is crucial for solving problems where the task is to determine how many different ways an event can happen.
An important tool in combinatorics is the combination formula, which tells us how many ways we can choose a subset of items from a larger set when the order is not important. This particular form of combinatorial calculation is known as 'combinations', where we use the combination formula given by:\[C(n, r) = \frac{n!}{r!(n-r)!}\]Here:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items we want to select.
• Factorials are vital to compute combinations efficiently.

In the exercise, when you are selecting 2 people from a group of 20, you are employing combinatorics to determine all possible pairs without regard to order.

Selection Without Order

Selection without order refers to choosing items from a set where the order in which the items are selected does not matter. In real-world scenarios, this is applicable in situations like forming teams or groups where the identity of members matters but the position does not.
For example, if you have a group of 20 people and you need to select 2 for a project, it doesn't matter if you choose person A first and person B second or vice versa. What matters is the pair of people you have chosen, not the sequence in which you picked them.
Combinations are used to solve problems of selection without order. This ensures that each possible selection is considered equivalent, regardless of the order in which it appears. By applying the formula for combinations, you can confidently determine how many unique groups or subsets can be formed, as seen in our example exercise which resulted in 190 distinct selections of 2 people from 20.