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17Calculus Precalculus - Factorials

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Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Factorials are just a way to write the multiplication of positive integers using special notation, since multiplication of sequential integers shows up a lot in calculus.
On this page, we cover the basics of factorials of numbers. On the next page, we extend this discussion to factorials involving variables.

Introduction To Factorials

For example, the idea is to write something like \( 1 \cdot 2 \cdot 3 \cdot 4 \) using special, more compact notation. Many times \(1\) is not included in the multiplication since it doesn't add anything, i.e. \( 1 \cdot 2 \cdot 3 \cdot 4 = 2 \cdot 3 \cdot 4 \).

We write this as \(1 \cdot 2 \cdot 3 \cdot 4 = 4!\), i.e. we multiply all positive integers starting with \(1\) up to and including whatever number appears with the exclamation point. So,
\(5! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5\).
Since multiplication is commutative, we can also write the numbers in reverse order, like this
\(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\).
The second way is sometimes easier to write since you start with the number with the exclamation point and go down.

Okay, let's work some practice problems.

Practice

Instructions - Unless otherwise instructed, determine the number these factorials represent.

\(7!\)

Problem Statement

Determine the number \(7!\) represents.

Solution

MIP4U - 1525 video solution

video by MIP4U

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\(4(5!)\)

Problem Statement

Determine the number \(4(5!)\) represents.

Solution

PatrickJMT - 1523 video solution

video by PatrickJMT

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\(\displaystyle{\frac{5}{6!}}\)

Problem Statement

Determine the number \(\displaystyle{\frac{5}{6!}}\) represents.

Final Answer

\(\displaystyle{\frac{5}{6!} = \frac{1}{144}}\)

Problem Statement

Determine the number \(\displaystyle{\frac{5}{6!}}\) represents.

Solution

\(\begin{array}{rcl} \displaystyle{\frac{5}{6!}} & = & \displaystyle{\frac{5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}} \\ & = & \displaystyle{\frac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 6}} \\ & = & \displaystyle{\frac{1}{144}} \end{array} \)

Final Answer

\(\displaystyle{\frac{5}{6!} = \frac{1}{144}}\)

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\(\displaystyle{6!-\frac{(7-3)!}{3!}}\)

Problem Statement

Determine the number \(\displaystyle{6!-\frac{(7-3)!}{3!}}\) represents.

Solution

PatrickJMT - 1524 video solution

video by PatrickJMT

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Building Factorials

The beauty of this, and something that will be important in calculus, is that factorials build on one another. So
\( \begin{array}{rcl} 5! & = & 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \\ & = & 5 \cdot ( 4 \cdot 3 \cdot 2 \cdot 1) \\ & = & 5 \cdot (4!) \end{array} \)
This is important since in calculus we will often have fractions with factorials in both the numerator and denominator and we need to cancel terms to simplify. Let's do an example.

Example 1

Simplify \(5! / 4! \)

\(\displaystyle{ \frac{5!}{4!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{ 4 \cdot 3 \cdot 2 \cdot 1} = 5 }\)

Notice that the terms, \( 1, 2, 3, 4 \) in both the numerator and denominator cancel leaving \(5\) in the numerator and \(1\) in the denominator.

PatrickJMT - Factorials - Evaluating Factorials! Basic Info [8min-34secs]

video by PatrickJMT

Okay, let's practice this concept by solving some problems.

Practice

Unless otherwise instructed, simplify these factorials.

\(\displaystyle{\frac{8!}{6!}}\)

Problem Statement

\(\displaystyle{\frac{8!}{6!}}\)

Solution

MIP4U - 1526 video solution

video by MIP4U

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\(\displaystyle{\frac{10!}{7!3!}}\)

Problem Statement

\(\displaystyle{\frac{10!}{7!3!}}\)

Solution

MIP4U - 1527 video solution

video by MIP4U

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\(\displaystyle{\frac{11!}{8!}}\)

Problem Statement

\(\displaystyle{\frac{11!}{8!}}\)

Solution

kbrescher - 1530 video solution

video by kbrescher

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\(\displaystyle{\frac{10!}{8!2!}}\)

Problem Statement

\(\displaystyle{\frac{10!}{8!2!}}\)

Solution

kbrescher - 1531 video solution

video by kbrescher

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\(\displaystyle{\frac{20!}{18!}}\)

Problem Statement

\(\displaystyle{\frac{20!}{18!}}\)

Final Answer

\(\displaystyle{\frac{20!}{18!} = 380}\)

Problem Statement

\(\displaystyle{\frac{20!}{18!}}\)

Solution

\(\begin{array}{rcl} \displaystyle{\frac{20!}{18!}} & = & \displaystyle{\frac{18! \cdot 19 \cdot 20}{18!} } \\ & = & 19 \cdot 20 = 380 \end{array} \)

Final Answer

\(\displaystyle{\frac{20!}{18!} = 380}\)

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\(\displaystyle{\frac{13!}{16!}}\)

Problem Statement

\(\displaystyle{\frac{13!}{16!}}\)

Final Answer

\(\displaystyle{\frac{13!}{16!} = \frac{1}{3360}}\)

Problem Statement

\(\displaystyle{\frac{13!}{16!}}\)

Solution

\(\begin{array}{rcl} \displaystyle{\frac{13!}{16!}} & = & \displaystyle{\frac{13!}{13! \cdot 14 \cdot 15 \cdot 16}} \\ & = & \displaystyle{\frac{1}{14 \cdot 15 \cdot 16} = \frac{1}{3360}} \end{array} \)

Final Answer

\(\displaystyle{\frac{13!}{16!} = \frac{1}{3360}}\)

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\(\displaystyle{\frac{23!}{5!20!}}\)

Problem Statement

\(\displaystyle{\frac{23!}{5!20!}}\)

Final Answer

\(\displaystyle{\frac{23!}{5!20!} = \frac{1771}{20}}\)

Problem Statement

\(\displaystyle{\frac{23!}{5!20!}}\)

Solution

\(\begin{array}{rcl} \displaystyle{\frac{!23}{5!20!}} & = & \displaystyle{\frac{20! \cdot 21 \cdot 22 \cdot 23}{20! \cdot 2 \cdot 3 \cdot 4 \cdot 5}} \\ & = & \displaystyle{\frac{7 \cdot 11 \cdot 23}{4 \cdot 5}} \\ & = & \displaystyle{\frac{1771}{20}} \end{array} \)

Final Answer

\(\displaystyle{\frac{23!}{5!20!} = \frac{1771}{20}}\)

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\(\displaystyle{\frac{(7+2)!}{6!2!}}\)

Problem Statement

\(\displaystyle{\frac{(7+2)!}{6!2!}}\)

Final Answer

\(\displaystyle{\frac{(7+2)!}{6!2!} = 252}\)

Problem Statement

\(\displaystyle{\frac{(7+2)!}{6!2!}}\)

Solution

\(\begin{array}{rcl} \displaystyle{\frac{(7+2)!}{6!2!}} & = & \displaystyle{\frac{9!}{6!2!}} \\ & = & \displaystyle{\frac{6! \cdot 7 \cdot 8 \cdot 9}{6! \cdot 2}} \\ & = & \displaystyle{\frac{7 \cdot 8 \cdot 9}{2}} \\ & = & \displaystyle{\frac{7 \cdot 4 \cdot 9}{1} = 252} \end{array} \)

Final Answer

\(\displaystyle{\frac{(7+2)!}{6!2!} = 252}\)

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\(\displaystyle{\frac{5!(4-3)!}{4!}}\)

Problem Statement

\(\displaystyle{\frac{5!(4-3)!}{4!}}\)

Final Answer

\(\displaystyle{\frac{5!(4-3)!}{4!} = 5}\)

Problem Statement

\(\displaystyle{\frac{5!(4-3)!}{4!}}\)

Solution

\(\begin{array}{rcl} \displaystyle{\frac{5!(4-3)!}{4!}} & = & \displaystyle{\frac{4! \cdot 5 \cdot 1!}{4!}} \\ & = & 5 \cdot 1 = 5 \end{array} \)

Final Answer

\(\displaystyle{\frac{5!(4-3)!}{4!} = 5}\)

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\(\displaystyle{ \frac{3! 7!}{6! 4!} }\)

Problem Statement

\(\displaystyle{ \frac{3! 7!}{6! 4!} }\)

Final Answer

\( 7/4 \)

Problem Statement

\(\displaystyle{ \frac{3! 7!}{6! 4!} }\)

Solution

2752 video solution

Final Answer

\( 7/4 \)

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\(\displaystyle{ \frac{12! 7!}{9! 6!} }\)

Problem Statement

\(\displaystyle{ \frac{12! 7!}{9! 6!} }\)

Final Answer

\( 12 \cdot 11 \cdot 10 \cdot 7 = 9240 \)

Problem Statement

\(\displaystyle{ \frac{12! 7!}{9! 6!} }\)

Solution

2753 video solution

Final Answer

\( 12 \cdot 11 \cdot 10 \cdot 7 = 9240 \)

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\(\displaystyle{ \frac{10! 6!}{8! 7!} }\)

Problem Statement

\(\displaystyle{ \frac{10! 6!}{8! 7!} }\)

Final Answer

\( 90/7 \)

Problem Statement

\(\displaystyle{ \frac{10! 6!}{8! 7!} }\)

Solution

2754 video solution

Final Answer

\( 90/7 \)

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Really UNDERSTAND Precalculus

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