Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to evaluate each expression. $$_{20} C_{5}$$

Step-by-Step Solution

Answer

Using a calculator, the expression \(_{20}C_5\) evaluates to 15,504.

Step 1: Understand the Combination Formula

The formula for a combination is given by the expression \( _nC_r = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, \( r \) is the number of items to choose, and \(!\) denotes factorial which means the product of all positive integers up to that number.

Step 2: Identify the Values of n and r

From the problem, \( n = 20 \) and \( r = 5 \). These are the values we will plug into the combination formula to evaluate.

Step 3: Calculate Factorials

Calculate \( 20! \), \( 5! \), and \( (20-5)! = 15! \). Since factorial numbers can be quite large, use a calculator to find:- \( 20! = 2432902008176640000 \)- \( 5! = 120 \)- \( 15! = 1307674368000 \)

Step 4: Plug Values into the Combination Formula

Insert the calculated factorials into the formula: \[_{20}C_5 = \frac{20!}{5! \times 15!} = \frac{2432902008176640000}{120 \times 1307674368000}\] This simplifies to: \[\frac{2432902008176640000}{156920924160000}\]

Step 5: Simplify the Fraction

Simplify the fraction using a calculator by performing the division:\[_{20}C_5 = \frac{2432902008176640000}{156920924160000} = 15504\]

Step 6: Confirm the Result

Make sure to double-check the calculations in each step using a calculator to ensure accuracy. The expression \( _{20}C_5 \) evaluates to 15,504.

Key Concepts

Factorial

In mathematics, factorial is a crucial concept often represented by the symbol "!".
This symbol denotes the product of all positive integers from 1 up to a specified number. For example, the factorial of 5, written as 5!, is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials are widely used in permutations, combinations, and other mathematical computations. The factorial of 0 is a special case, defined to be 1.
This special case is due to its significance in various formulas, ensuring they work correctly even when counting down to zero. Calculating larger factorials, like 20!, involves multiplying a large sequence of numbers, which can result in very large values.
That's why using a calculator or software to compute these numbers is often necessary. For example:

• 20! = 2,432,902,008,176,640,000

Combination Formula

The combination formula is used in combinatorics to determine the number of ways you can select items from a larger set.
This is typically represented as \(_nC_r\), where \(n\) is the total number of items, and \(r\) is the number of items you choose. The formula is written as:\[ _nC_r = \frac{n!}{r!(n-r)!} \]Understanding this formula is pivotal for solving problems involving groups or selections.
Unlike permutations, combinations do not consider the order of selection.For instance, if you have 20 fruits and you want to select 5,
the number of ways to do so is found using:

• \(\_{20}C_5 = \frac{20!}{5! \times 15!}\)

The "!" symbol once again indicates a factorial, meaning you're dealing with products of sequences of integers.

Calculator Usage

While solving mathematical problems like combinations, using a calculator becomes essential.
This is especially true when dealing with large numbers in factorial calculations.When using a calculator for factorials and combinations, follow these steps:

• Access the factorial function, often located under advanced or scientific mode.
• Calculate the factorials needed separately (like \(20!\) and \(5!\) for \(_{20}C_5\)).
• Perform operations using the combination formula.
• Divide large numbers carefully to avoid mistakes.

Many calculators come with a direct combination or "nCr" function to simplify calculations.
Entering \(n\) and \(r\), the calculator provides the result directly.
Ensure you double-check each entry and operation by repeating calculations if unsure.

Problem-Solving Steps

Solving problems involving combinations requires careful attention to detail.
The following sequential steps can guide you through solving expressions like \(_{20}C_5\):

• Understanding the formula: Grasp the combination formula \(_nC_r = \frac{n!}{r!(n-r)!}\) and its components.
• Identify your variables: From the problem, figure out values for \(n\) and \(r\). For example, \(n = 20\) and \(r = 5\).
• Calculate factorials: Use a calculator to find \(n!\), \(r!\), and \((n-r)!\).
• Apply the formula: Plug these values into the combination formula.
• Simplify the result: Use the calculator to divide factorial products and simplify.
• Confirm your solution: Double-check every step for accuracy, especially when dealing with large numbers.

Adhering to these problem-solving steps can minimize errors and help in understanding the process better.
This method not only leads to correct answers but also improves problem-solving skills in mathematics.