Problem 90 - Algebra and Trigonometry

Question

Evaluate \(\frac{n !}{(n-r) !}\) for \(n=20\) and \(r=3\)

Step-by-Step Solution

Answer

The result of the calculation is \(20 × 19 × 18 = 6840\).

Step 1: Determine \(n!\)

First, we need to calculate \(n!\). The factorial is defined as the product of an integer and all the integers below it. Thus, \(20! = 20 × 19 × 18 × ... × 3 × 2 × 1\).

Step 2: Determine \((n-r)!\)

Next, we calculate \((n-r)!\), which means we subtract \(r=3\) from \(n=20\) to get \(17\), and then find \(17!\). Thus, \(17! = 17 × 16 × 15 × ... × 3 × 2 × 1\).

Step 3: Perform the Division

Now, we perform the division operation. The division \(\frac{20!}{17!}\) will basically cancel out all the terms from \(17\) down to \(1\) in both the numerator and the denominator, leaving \(20 × 19 × 18\).

Key Concepts

Factorial Calculation

Factorial calculation may sound complex at first, but it's a straightforward concept in mathematics. A factorial of a non-negative integer, denoted by an exclamation mark (!), is the product of that integer and all the positive integers less than it. To put it simply, to find the factorial of a number, you multiply it by the number that comes right before it, then multiply the result by the next number down, and so on, until you reach one.

For example:

• The factorial of 5, written as 5!, is calculated as: \(5! = 5 imes 4 imes 3 imes 2 imes 1 = 120\)
• Similarly, 4! is calculated as \(4! = 4 imes 3 imes 2 imes 1 = 24\)

Factorials grow very large, very quickly. This is why they are often used in calculations involving large numbers of combinations and permutations.

In our exercise, calculating the factorial is the first step in evaluating expressions like \( \frac{20!}{17!} \).

Combinatorial Mathematics

Combinatorial mathematics involves counting, arrangement, and combination principles that are fundamental in understanding how elements from a set can be selected or organized. In these problems, factorial notation is key due to its ability to handle large numbers and intricate permutations.

The permutations of a set refer to all possible ways in which the elements of the set can be ordered. For example, if you have three items: A, B, and C, permutations would show how you can arrange these items, such as ABC, ACB, BAC, etc.

Combining factorials in combinatorial problems helps simplify expressing permutations and combinations. For instance, to find the permutations of arranging 'r' items from 'n' total items, we use the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]
This formula stems from the rationale that for an arrangement of 'r' items, we need the first 'r' factors of the factorial of all 'n' items. Understanding concepts like these in combinatorial mathematics is essential for problems involving selections and arrangements.

Division of Factorials

Division of factorials might sound tricky, but understanding it can be quite enlightening. It often appears in problems where you want to simplify the factorial notation that involves subtraction of terms.

When dividing two factorial expressions, such as \( \frac{n!}{(n-r)!} \), we can cancel out terms that appear in both the numerator and the denominator. In our original exercise, where n equals 20 and r equals 3, we calculate: \[ \frac{20!}{17!} \]
Here's how it simplifies:

• \(20!\) means multiplying all integers from 20 down to 1.
• \(17!\) means multiplying integers from 17 down to 1.
• Dividing \(20!\) by \(17!\) removes the common terms from both the top and bottom, resulting in just \(20 \times 19 \times 18\) .

This representation gives you the product of 'r' consecutive numbers starting from 'n', which is crucial in simplifying complex combinatorial expressions. It's an effective way of dealing with large numbers, as fully calculating factorials can be cumbersome.