Problem 4 - A Graphical Approach to College Algebra

Question

Five slips of paper, each of which is marked with the number \(1,2,3,4,\) or \(5,\) are placed in a box. After mixing well, two slips are drawn, with the order not important.

Step-by-Step Solution

Answer

There are 10 ways to draw two slips from the box.

Step 1: Determine the Total Number of Slips

There are five slips of paper marked with the numbers 1, 2, 3, 4, and 5. This means we have a total of 5 slips in the box.

Step 2: Choose Two Slips

Since order is not important, we need to find the number of ways to choose 2 slips from a total of 5 slips. This is a combination problem, which can be solved using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). In this case, \( n = 5 \) and \( r = 2 \).

Step 3: Calculate the Number of Combinations

Substitute \( n = 5 \) and \( r = 2 \) into the combination formula: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10. \] So, there are 10 different ways to choose 2 slips from 5.

Key Concepts

Understanding Probability

Probability is a fundamental concept in mathematics that measures the likelihood of an event occurring. It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. A probability of 0.5 suggests that an event is equally likely to occur or not occur.

A key formula for probability is given by the ratio:

• Probability of an event (E) = \( \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

In our exercise, if you want to find the probability of drawing any two specific slips, you would calculate the number of favorable outcomes (ways to draw those slips) divided by the total number of ways to draw any two slips (10 as found from the combination calculation).

This understanding helps in determining how likely it is that a particular event will happen, which is an essential part of probability theory used in many fields such as statistics, finance, and science.

Exploring Combinatorics

Combinatorics is a branch of mathematics that focuses on counting and arranging items. It gives us tools to answer questions such as 'How many different ways can a set of items be arranged?' or 'In how many ways can we choose a subset of items from a larger set?'.

In our exercise, the main question is how to choose 2 slips from a total of 5. This is a typical combinatorial problem where the order does not matter. We use combinations to solve these kinds of problems because they count the number of ways to choose items without regard to their arrangement. The formula for combinations is:

• Combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Thus, combinatorics helps us efficiently determine the count of different possible selections or arrangements, a crucial skill in solving many real-world problems.

Delving into the Binomial Coefficient

The binomial coefficient is a pivotal part of combinatorial mathematics. It is symbolized by \( \binom{n}{r} \) and is explicitly used to determine the number of ways to choose \( r \) objects from \( n \) objects, independent of order.

You might often hear it referred to as "n choose r." The formula to calculate it is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \(!\) signifies factorial, which means you multiply a number by each positive integer less than it (e.g., \(5! = 5 \times 4 \times 3 \times 2 \times 1\)).

In our scenario, by applying the formula \( \binom{5}{2} \), you find there are 10 ways to select two slips of paper from five. This calculation is critical in combinatorics and simplifies problems across various applications, such as game theory, genetics, and computer science.