Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to evaluate each expression. \({ }_{32} C_{4}\)

Step-by-Step Solution

Answer

The result of \( _{32}C_4 \) is 35,960.

Step 1: Understand the Combination Formula

The combination formula is used to determine the number of ways to choose a subset of items from a larger set, without regard to the order of selection. It is given by the expression \[ _nC_k = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, and \( k \) is the number of items to choose. In this problem, \( n = 32 \) and \( k = 4 \).

Step 2: Calculate the Factorials

We need to calculate the factorials in the formula: \( n! = 32! \), \( k! = 4! \), and \( (n-k)! = 28! \). Factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \).

Step 3: Plug Values into the Formula

Now substitute the calculated factorials into the combination formula:\[ _{32}C_4 = \frac{32!}{4!\cdot 28!} \]

Step 4: Simplify the Expression Using a Calculator

Using a calculator, compute the expression. We note that \( 32! \) over \( 28! \) simplifies as \( 32 \times 31 \times 30 \times 29 \), because the terms in \( 28! \) cancel out, leaving us with the expression: \[ \frac{32 \times 31 \times 30 \times 29}{4!} = \frac{32 \times 31 \times 30 \times 29}{24} \]

Step 5: Compute the Final Result

Using a calculator, evaluate the above expression:\[ _{32}C_4 = \frac{32 \times 31 \times 30 \times 29}{24} = 35,960 \]

Key Concepts

Factorials

A factorial is a mathematical operation that is used in many areas, including combinations and permutations. The factorial of a positive integer \( n \), denoted \( n! \), is the product of all positive integers up to \( n \).
For example:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 1! = 1 \) (by convention)
• \( 0! = 1 \) (also by convention)

Factorials grow very quickly as \( n \) increases, which is why calculating them manually for larger numbers like \( 32! \) can be laborious. Many calculators have a factorial button that simplifies this process greatly, allowing for these calculations to be quick and error-free.

Combination Formula

The combination formula is a useful tool in probability and statistics to determine how many different groups or subsets can be formed from a larger set of distinct items. It is given by the expression
\[ _nC_k = \frac{n!}{k!(n-k)!} \] where:

• \( n \) is the total number of items.
• \( k \) is the number of items to choose.

The role of the combination formula is to calculate how many ways you can pick \( k \) items from \( n \) without caring about the order. In this exercise, we're tasked with finding \(_{32}C_{4}\), which means choosing 4 items from a set of 32.
This formula ensures that there is no repetition, and the order of selection does not matter, focusing solely on the unique combinations possible.

Calculator Use

Using calculators in combination problems streamlines the process and avoids errors in manual calculations, especially when dealing with large numbers. To evaluate a combination like \({ }_{32}C_{4}\), you should first understand that you will need to compute factorials where needed.
A good calculator will have at least a factorial function, often labeled as "n!". Here’s how you can simplify using a calculator:

• Calculate \( 32! \) and \( 28! \). Once you have those, you can focus on the terms \( 32 \times 31 \times 30 \times 29 \) by realizing that \( \frac{32!}{28!} \) equals that product.
• Input these numbers directly into the calculator to find \( 4! \), which is 24.
• Finally, compute \( \frac{32 \times 31 \times 30 \times 29}{24} \) to find \( _{32}C_4 = 35,960 \).

While calculators simplify the process, a solid grasp of the underlying formulas and logic is crucial for efficient and error-free calculations.