Problem 85 - Algebra and Trigonometry

Question

use a calculator’s factorial key to evaluate each expression. $$ \frac{54 !}{(54-3) ! 3 !} $$

Step-by-Step Solution

Answer

After carefully entering these calculations into the calculator, this will give the final result of the evaluated expression.

Step 1: Identify each expression

The expression \(\frac{54 !}{(54-3) ! 3 !}\) has three parts that use the factorial function. The first part, \(54 ! \), the second, \((54-3) !\), and the last, \(3 !\).

Step 2: Calculate factored expressions

First, individually calculate the factorial of each of these numbers using the factorial function on your calculator. Therefore, \(54!\), \((54 - 3)!\) or \(51!\), and \(3!\) are evaluated.

Step 3: Compute the final result

Lastly, divide 54! by the result of multiplying \(51 !\) and \(3 !\). Carefully entering these calculations into the calculator will give the final result.

Key Concepts

Factorial Function

The factorial function, written as \(n!\), is a fundamental concept in combinatorics and mathematics. It represents the product of all positive integers from 1 up to \(n\). This means that for any given number, its factorial is: \[ n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1 \] For example, \(5!\) or "five factorial" is computed as: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] The factorial function grows very quickly with larger numbers, making its manual computation cumbersome. That’s where calculators with a factorial function key come in handy, as they can compute such large numbers effortlessly.

Calculator Usage in Mathematics

Calculators are powerful tools in mathematics, especially for operations involving large numbers like factorials. Many scientific calculators include a factorial key, often labeled as \(!\), which simplifies the computation. Here's how to use it:

• Enter the number you wish to compute the factorial of.
• Press the factorial key \(!\), which is sometimes found by pressing the "shift" or "2nd" function button first.
• The calculator will display the result instantly.

This method is efficient and greatly reduces errors that may occur with manual calculations. It is important to ensure you enter the numbers correctly, particularly when dealing with expressions involving multiple factorials like in our example.

Evaluating Expressions

Evaluating mathematical expressions involves a systematic approach to compute their value. In our example, we use the expression: \(\frac{54 !}{(54-3) ! 3 !}\). The expression consists of both a numerator and a denominator. It is crucial to handle each part correctly:

• First, calculate the factorial of each component separately, in this case, \(54!\), \((54-3)!\), and \(3!\).
• Next, multiply the results of the factorials in the denominator \((51! \times 3!)\).
• Finally, divide the factorial value of the numerator \(54!\) by the product obtained from the denominator \((51! \times 3!)\).

These steps ensure the expression is evaluated accurately and efficiently.

Binomial Coefficient

The expression \(\frac{54 !}{(54-3) ! 3 !}\) is known as a binomial coefficient. Binomial coefficients, denoted by \(\binom{n}{k}\), appear in combinatorics when calculating combinations. They represent the number of ways to choose \(k\) items from a set of \(n\) items. The formula is: \[ \binom{n}{k} = \frac{n!}{(n-k)! \, k!} \] In our example, \(\binom{54}{3}\) tells us how many ways we can select 3 items from a set of 54. The binomial coefficient is foundational in probability, algebra, and statistics, providing insights into possible outcomes of selections.