Problem 47 - College Algebra

Question

Use the formula for \(_{n} C\), to solve Exercises \(41-48\). To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Step-by-Step Solution

Answer

The number of different selections possible is 22,957,480.

Step 1: Identify n and r

Firstly, identify the values for n and r. In this case, n equals 53 because there are 53 numbers to begin with, and we're selecting 6 numbers, so r equals 6.

Step 2: Apply Combination Formula

Using the combination formula, we can plug in the values of n and r to get \(_{53} C_{6}= \frac{53!}{6!(53-6)!}\).

Step 3: Solve the Equation

Now, proceed to solve this equation. Factorial means to multiply a series of descending natural numbers, so 53! means multiplying all natural numbers from 53 to 1. But as the combination formula contains a subtraction of factorials, using this property gives an easier calculation path 'n! = n * (n-1)!', so the calcualtion can be simplified as ' \(_{53} C_{6}= \frac{53*52*51*50*49*48}{6*5*4*3*2*1}\). After performing the calculation, the solution is 22,957,480.

Step 4: Interpret the Result

The result tells us that there are 22,957,480 unique ways to select 6 numbers from a pool of 53, regardless of their order. Therefore, the chances of winning the LOTTO in Florida by correctly selecting these 6 numbers are 1 out of 22,957,480.

Key Concepts

Combinations

In mathematics, combinations refer to a way of selecting items from a collection, such that the order of selection does not matter. This concept is essential in scenarios where you need to choose a specific number of elements from a larger set without regard to the sequence. For example, when picking your lottery numbers, it doesn't matter in which order you choose them. The formula for combinations is denoted by \[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \]where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to choose.
• \(!\) represents a factorial operation.

Combinations are used in diverse fields, including statistics, mathematics, finance, and gaming scenarios like lotteries.

Factorials

Factorials are a fundamental concept in combinatorics, essential for calculating permutations and combinations. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For instance, \( 5! \) equals \( 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \). By convention, \( 0! \) is 1.Here are some key features of factorials:

• Grow very large very quickly as \( n \) increases.
• Used to simplify calculations in probability and statistics.
• A building block for understanding complex mathematical operations.

Learning factorials is crucial because they help calculate the number of ways to arrange a set of items, playing a vital role in solving combination and permutation problems.

Lottery Probability

Lottery probability is the branch of mathematics that deals with the likelihood of winning in a lottery game. The concept is rooted in combinations because it involves selecting a subset of numbers from a larger pool. For the Florida LOTTO, you select 6 numbers from a total of 53. The formula for calculating the odds of winning can be represented using combinations:\[ \text{Probability of winning} = \frac{1}{_{n}C_{r}} \]In Florida's case, it becomes:\[ \frac{1}{_{53}C_{6}} = \frac{1}{22,957,480} \]which implies you have one chance in 22,957,480.Some important points about lottery probabilities:

• They help understand how difficult it is to win.
• Combinations minimize "order matters" concerns, focusing purely on selection.
• Understanding these probabilities can keep gambling expectations realistic.

Mathematical Problem-Solving

Mathematical problem-solving involves using mathematical concepts and methods to figure out solutions to various issues. In solving the Florida LOTTO problem, key mathematical skills were applied:

• Identifying what the question asks: Here, it was about finding the total number of possible combinations.
• Recognizing the correct formula: Use of \(_{n}C_{r}\) for non-order-specific selection problems.
• Performing calculations: Breaking down the equation details and simplifying using factorials.
• Interpreting results: Understanding the significance of the final number in a real-world context.

This approach to problem-solving not only provides answers but also develops critical thinking skills. Mathematical challenges connect theoretical concepts with practical applications, enhancing overall strategic thinking and analytical capabilities.