Problem 52 - A Graphical Approach to College Algebra

Question

\text {Solve each problem involving combinations.} Apple Samples How many different samples of 3 apples can be drawn from a crate of 25 apples?

Step-by-Step Solution

Answer

2300 different samples of 3 apples can be drawn.

Step 1: Understand the Problem

We need to determine how many different groups of 3 apples can be drawn from a total of 25 apples. This is a problem involving combinations since the order in which we select the apples does not matter.

Step 2: Identify the Formula for Combinations

The formula for finding the number of combinations (denoted as \( C(n, r) \)) is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and "!" denotes the factorial, which is the product of all positive integers up to that number.

Step 3: Substitute Values into the Formula

For this problem, \( n = 25 \) and \( r = 3 \). Substitute these values into the combination formula:\[ C(25, 3) = \frac{25!}{3!(25-3)!} \]

Step 4: Simplify the Factorials

Calculate the factorials involved:- \( 25! = 25 \times 24 \times 23 \times 22! \)... (which can be left as is since we will cancel out)- \( 3! = 3 \times 2 \times 1 = 6 \)- \( (25-3)! = 22! \)Substitute these into the combination expression:\[ C(25, 3) = \frac{25 \times 24 \times 23 \times 22!}{6 \times 22!} \]Cancel out \( 22! \) in the numerator and the denominator:

Step 5: Calculate the Combinations

Continue simplifying:\[ C(25, 3) = \frac{25 \times 24 \times 23}{6} \]Now, multiply the top row and divide by 6:- \( 25 \times 24 = 600 \)- \( 600 \times 23 = 13800 \)- \( \frac{13800}{6} = 2300 \)

Step 6: Conclusion

Thus, the number of different samples of 3 apples that can be drawn from a crate of 25 apples is \( 2300 \).

Key Concepts

Combinations

Combinations are a way of selecting items from a larger pool where the order does not matter. This is a key point in many mathematical problems, as we are often interested in the group as a whole, rather than the sequence the items are selected.
For instance, if we have 25 apples and want to select groups of 3, we do not care about which apple is picked first, second, or third; only which apples make up that group of 3.
To calculate combinations, we use a specific formula: \( C(n, r) = \frac{n!}{r!(n-r)!} \). If you see problems that revolve around choosing groups where sequence doesn’t matter, think combinations!

• Choosing a team of players from a larger group
• Selecting a set number of ingredients from a list for a recipe
• Drawing specific cards from a deck

This makes combinations a powerful tool in analytical and real-world problem solving.

Factorial

The factorial function is an important operation in mathematics, especially in combinatorics.
Represented by an exclamation mark (!), a factorial is the result of multiplying a sequence of descending positive numbers from any given number down to 1. So, \( n! \) (read as "n factorial") means \( n \times (n - 1) \times (n - 2) \times ... \times 1 \).
Factorials help simplify calculations in permutations and combinations.

• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Factorials grow rapidly, which is why using them in combination gives us a way to make sense of large numbers without overcomplication.

Understanding how factorials cancel out in equations, like in combinations, helps in breaking down and solving complex problems more easily.

Mathematical Problem Solving

Solving mathematical problems often requires a clear and methodical approach.
A typical process involves:

• Understanding the question thoroughly to determine what is being asked.
• Identifying the right concepts and formulas to apply, such as combinations in the apple problem.
• Substituting the given numbers into the correct formula.
• Simplifying the calculations step by step, using mathematical operations such as factorials and division.

This logical progression helps in solving not just textbook problems, but any real-world scenarios where mathematics is applied. Practice is key to mastering problem-solving skills, allowing you to recognize patterns and apply the correct strategies efficiently.

Permutation

Permutations are different from combinations because they consider the order of selection as important.
While combinations deal with groups irrespective of their arrangement, permutations focus on every possible way the items can be ordered.For example, if you were arranging 3 different apples on a shelf, permutations would count every unique arrangement.
The number of permutations of \( r \) items from \( n \) total items is given by the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]Unlike combinations, this is about sequence and distinctive ordering.

• Think of arranging books on a shelf: swapping the books' positions changes the permutation.
• Arranging people in a line where the order they stand in matters.

Permutations are crucial when the sequence is important, providing a complete count of all unique arrangements.