Problem 58 - Precalculus with Limits

Question

In Exercises 57 - 60, evaluate \( _nC_r \) using a graphing utility. \( _{10}C_7 \)

Step-by-Step Solution

Answer

The value of \( _{10}C_7 = 120 \)

Step 1: Identify values of n and r

In the combination notation \( _{10}C_7 \), here '10' is the total number of items 'n' and '7' is the number of items to choose 'r'.

Step 2: Apply the Combination Formula

The formula for combination is \( _nC_r = \frac{n!}{r!(n-r)!} \). Applying the values, we get \( _{10}C_7 = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} \)

Step 3: Calculate Factorials

Calculate the factorials in the formula. The factorial of a number 'n' (denoted as n!) is the product of all positive integers less than or equal to 'n'. \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \), \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \), and \( 3! = 3 \times 2 \times 1 \)

Step 4: Solve the Expression

On solving the expression \( _{10}C_7 = \frac{10!}{7!3!} \), we get \( _{10}C_7 = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \)

Key Concepts

Understanding Factorials

In mathematics, a factorial is a function that multiplies a number by every number below it until it reaches one. For example, the factorial of 5, denoted as 5!, is equal to \(5 \times 4 \times 3 \times 2 \times 1\). This operation is essential when working with combinations because it calculates the total number of ways to organize a set of objects.

Factorials are represented by the exclamation point symbol "!" following a number. Here are a few properties of factorials that are useful to remember:

• 0! is equal to 1: This might seem counterintuitive, but defining 0! as 1 maintains the consistency of various mathematical formulas.
• The factorial function grows very quickly with larger values of n.
• Factorials are only defined for non-negative integers.

When solving problems involving combinations, you'll often work with expressions like \( n!/(r!(n-r)!)\), where you'll need to calculate the factorials first.

Using the Combination Formula

The combination formula is a pivotal concept for determining how many ways you can choose a subset of items from a larger set, without considering the order. The formula for combinations is \( _nC_r = \frac{n!}{r!(n-r)!} \). Here, \( n \) is the total number of items and \( r \) is the number of items to choose.

Let’s break this down using our example \( _{10}C_7 \):

• Identify \( n \) and \( r \): In \( _{10}C_7 \), n is 10, and r is 7.
• Apply the formula: Plug in the values into the formula \( _{10}C_7 = \frac{10!}{7!3!} \).
• Solve the factorials: Calculate each factorial involved (like \( 10! \), \( 7! \), and \( 3! \)).
• Calculate the result: Simplify the expression to find the number of combinations, in this case, 120.

By using the combination formula, you're efficiently calculating the total number of ways to choose 7 items from a set of 10, ensuring you don't count permutations where the order would matter.

Leveraging a Graphing Utility

Graphing utilities can greatly simplify the computation of combinations, especially when dealing with larger numbers. These tools, like calculators or computer software, have built-in functions for combinations that do all the heavy lifting for you.

Here’s how you can use a graphing utility to evaluate combinations:

• Input the combination function: Most graphing calculators have a "nCr" function which stands for combination. You simply input the values of \( n \) and \( r\).
• Navigate the menus: Often, these calculators require you to enter the total number of items (n) first, use the combination function button, and then enter the number of items to choose (r).
• Press calculate: Once your numbers are in place, simply press enter or the calculate button. The graphing utility will swiftly and accurately solve \( _{10}C_7 \) for you!

Using graphing utilities not only saves time but also reduces the likelihood of errors that can occur during manual computations.