Problem 34 - Thinking Mathematically

Question

The members of the Student Activity Council on your campus are meeting to select two speakers for a month-long event exploring why some people are most likely to succeed. The choices are Bill Gates, Oprah Winfrey, Mark Zuckerberg, Hillary Clinton, and Steph Curry. How many different ways can the two speakers be selected?

Step-by-Step Solution

Answer

There are 10 different ways the two speakers can be selected for the event.

Step 1: Understanding Combinations

Combinations refer to the selection of items from a larger set where the order of selection does not matter. In this scenario, we are looking for the combinations of 2 individuals who can be selected from a total of 5. This can be represented mathematically by the notation C(5,2) or 5C2.

Step 2: Formula for Combinations

The formula for combinations is given by: \[C(n, r) = \frac{n!}{r!(n-r)!}\], where 'n' represents the total number of items to select from (in this case 5), 'r' represents the number of items to select (in this case 2), and '!' indicates the factorial of a number, which essentially means multiplying that number by all positive integers less than it.

Step 3: Calculate Factorials

Firstly, calculate the factorials needed for the formula: \(5!= 5*4*3*2*1 = 120\), \(2!= 2*1 = 2\), and \((5-2)!= 3*2*1 = 6\).

Step 4: Substitute the values in formula

Substitute these values into the formula: \[C(5,2) = \frac{5!}{2!(5-2)!} = \frac{120}{2*6} = 10\]

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that helps us understand and solve problems related to counting and arrangements. It deals with how we can group or arrange various objects in sets or lists, particularly when the order doesn't matter. In the provided exercise, we are dealing with a problem where we need to select two speakers from a group of five candidates.

When we talk about combinations in combinatorics, like choosing 2 speakers from 5, the order of selection doesn't matter. This is different from permutations, where the order does matter. For example, choosing Bill and Oprah is the same as choosing Oprah and Bill, so they represent only one combination, not two different ones.

The notation for combinations, often written as C(n, r) or nCr, indicates the number of ways to choose r items from a set of n. It's an essential concept to grasp because it helps in solving a wide array of counting problems without exhausting all possibilities.

Factorials

Factorials might sound daunting at first, but they are simply products of all positive integers up to a given number. The factorial of a number 'n' is denoted by 'n!' and calculated as the product: n × (n - 1) × (n - 2) × ... × 1. Factorials are foundational to understanding various combinatorial problems, including calculating combinations.

In the original exercise, we need to compute the factorials of 5, 2, and 3 because the formula for combinations involves these values.

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 2! = 2 × 1 = 2
• 3! = 3 × 2 × 1 = 6

Factorials grow rapidly with larger numbers, and although they are simple to calculate for numbers like 5, they can become cumbersome with bigger numbers. Luckily, calculators and software can easily handle these operations.

Understanding how to compute and use factorials is crucial for effectively applying combinations, permutations, and other combinatorial calculations.

Mathematical Notation

Mathematical notation is a language that helps us succinctly express mathematical ideas and procedures. It's an essential tool for communicating complex concepts in a simple way. In the context of combinatorics, mathematical notation allows us to define combinations using the formula: \[C(n, r) = \frac{n!}{r!(n-r)!}\]

Here, 'n' represents the total number of items to choose from, while 'r' is the number of items we pick. The exclamation mark indicates factorial, which helps us manage the orderliness (or disorderliness!) by accounting for all the various ways these selections can occur.

Using mathematical notation is crucial because it allows mathematicians and students to communicate on a global scale. By representing complex operations compactly, mathematical notation enables problem solvers to focus on understanding and applying the concepts rather than getting bogged down in lengthy explanations or calculations. This notation also makes it simple to compute the number of combinations or configurations possible in any selection problem.