13. Counting & Probability

Factorials

13. Counting & Probability

Factorials: 동영상 및 연습문제

주제 요약

Factorials use an exclamation point to represent the product of all whole numbers from a given positive integer down to 1. A factorial can be written recursively as $n!=n(n-1)!$ , which shows that each new factorial is the previous factorial multiplied by one more number. Important base values include $0! = 1$ and $1!=1$ .

To simplify factorial expressions, expand only as much as needed rather than multiplying everything out. This is especially useful in a fraction, where rewriting the larger factorial to include a matching smaller factorial allows common factors to cancel. When factorials appear with division, cancellation works only after the expression is rewritten so the same factorial factor appears in both numerator and denominator. This idea makes large factorial expressions much easier to evaluate accurately.

개념

Factorials

영상 길이:

Factorials Video Summary

Factorials are a mathematical operation denoted by an exclamation point (!), representing the product of all whole numbers from a given number down to 1. For instance, the factorial of 4, written as 4!, is calculated as:

4! = 4 × 3 × 2 × 1 = 24

Understanding factorials is essential in various fields such as combinatorics, probability, and sequences. The factorial of 1 (1!) is simply 1, as there are no numbers to multiply below it. Similarly, 2! equals 2 (2 × 1), and 3! equals 6 (3 × 2 × 1). Continuing this pattern, we find:

3! = 6, 4! = 24, and 5! = 120.

A key observation is that each factorial can be expressed in terms of the previous factorial. For example, 6! can be calculated as:

6! = 6 × 5! = 6 × 120 = 720.

In general, for any integer n, the factorial can be defined recursively as:

n! = n × (n - 1)!

This recursive relationship allows for simplification of factorial expressions. For example, to simplify 4 × 3!, we can recognize that:

4 × 3! = 4! = 24.

Another example involves dividing factorials, such as 100! / 99!. Instead of calculating both factorials fully, we can use the recursive definition:

100! = 100 × 99!,

which simplifies the expression to:

100! / 99! = 100.

Additionally, the concept of 0! is often confusing. However, through the recursive definition, we can derive that:

1! = 1 × 0!,

and since 1! equals 1, it follows that:

1 = 1 × 0!,

leading us to conclude that:

0! = 1.

Understanding these principles of factorials not only aids in simplifying expressions but also enhances problem-solving skills in mathematics. Practice with various factorial problems will solidify this knowledge and improve computational efficiency.

문제

Evaluate the expression.
$\frac{12}{4!}$

$$\frac$14$

$$\frac$12$

$2$

$3$

문제

Evaluate the expression.
$\frac{9!}{7!}$

문제

Evaluate the expression.
$\frac{16!}{12! \cdot 4!}$

1820

43,680

학생들이 이 주제에 대해 묻는 질문은 다음과 같습니다:

A factorial, denoted by an exclamation point $!$ , is the product of all whole numbers from a given positive integer down to 1. For example, the factorial of 4, written as $4_{!}$ , is calculated as $4 imes 3 imes 2 imes 1 = 24$ . Factorials are used in various areas such as combinatorics, sequences, and probability. The factorial operation can also be defined recursively as $n_{!} = n imes n_{- 1!}$ , meaning each factorial is the current number multiplied by the factorial of the previous number. This recursive definition helps simplify calculations and expressions involving factorials.

The value of $0_{!}$ is defined as 1 by convention to maintain consistency in mathematical formulas, especially in combinatorics. This can be understood using the recursive factorial formula: $1_{!} = 1 imes 0_{!}$ . Since $1_{!} = 1$ , it follows that $0_{!} = 1$ . Defining $0_{!}$ as 1 ensures that factorial expressions and formulas work smoothly without exceptions, especially when calculating combinations or permutations where zero elements are chosen.

Factorial expressions can be simplified by using the recursive property of factorials and canceling common factors instead of fully expanding them. For example, consider the expression $\frac{100_{!}}{99_{!}}$ . Using the recursive definition, $100_{!} = 100 imes 99_{!}$ , so the expression becomes $\frac{100}{imes}$ . The $99_{!}$ terms cancel out, leaving just $100$ . This method avoids large multiplications and makes calculations more efficient, especially for large numbers.

The recursive formula for factorials is $n_{!} = n imes n_{- 1!}$ , where $n$ is a positive integer. This means the factorial of $n$ is the product of $n$ and the factorial of $n - 1$ . This formula helps simplify calculations by breaking down a factorial into smaller parts. For example, to find $6_{!}$ , you can calculate $6 imes 5_{!}$ . Since $5_{!}$ is often known or easier to compute, this reduces the complexity of the calculation. It also aids in simplifying factorial expressions by allowing cancellation of common factorial terms.

Factorials are fundamental in combinatorics and probability because they count the number of ways to arrange or select items. For example, the number of ways to arrange $n$ distinct objects is $n_{!}$ . In combinations and permutations, factorials help calculate the total possible outcomes. For instance, the number of ways to choose $k$ items from $n$ is given by the combination formula $\frac{n_{!}}{k_{!} imes n_{- k!}}$ . Simplifying these expressions often involves canceling factorial terms, making factorial knowledge essential for solving problems in these fields.