Factorial Notation and Its Operational Rules | 阶乘记号及其运算规则

📚 Factorial Notation and Its Operational Rules | 阶乘记号及其运算规则

The factorial is one of the most fundamental concepts in A-Level Mathematics, appearing in permutations, combinations, binomial expansions, and series. This article provides a comprehensive guide to factorial notation and its operational rules, tailored specifically for A-Level students.

阶乘是 A-Level 数学中最基本的概念之一，出现在排列、组合、二项式展开和数列等多个章节中。本文专门为 A-Level 学生量身打造，系统讲解阶乘记号及其运算规则。

1. Definition of Factorial | 阶乘的定义

For a positive integer n, the factorial of n, denoted as n!, is defined as the product of all positive integers from 1 to n. That is:

对于正整数 n，n 的阶乘记作 n!，定义为从 1 到 n 的所有正整数的乘积，即：

n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1

For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The factorial grows extremely quickly: 10! is already 3,628,800.

例如，5! = 5 × 4 × 3 × 2 × 1 = 120。阶乘的增长速度极快：10! 已经达到 3,628,800。

By convention, we define 0! = 1. This definition is not arbitrary; it ensures that formulas involving combinations and series remain consistent.

按照约定，我们定义 0! = 1。这个定义并非随意设定，而是为了确保组合数和级数中的公式保持一致。

2. Basic Properties | 基本性质

The most important property of factorials is the recursive relationship. For any positive integer n:

阶乘最重要的性质是递推关系。对于任意正整数 n：

n! = n × (n − 1)!

This property forms the backbone of many algebraic manipulations involving factorials. For instance, 8! = 8 × 7!, and more generally, we can express any factorial in terms of a smaller factorial.

这一性质构成了许多涉及阶乘的代数运算的基础。例如，8! = 8 × 7!，更一般地，我们可以用较小的阶乘来表示任意阶乘。

Another useful property is that factorials of consecutive integers differ by a multiplicative factor, not an additive one. This distinction is crucial when simplifying expressions.

另一个有用性质是：连续整数的阶乘之间相差一个乘法因子，而非加法因子。这一区别在化简表达式时至关重要。

3. Simplifying Factorial Expressions | 化简阶乘表达式

In A-Level examinations, students are frequently required to simplify expressions containing factorials. The key strategy is to expand the larger factorial partially to reveal common factors.

在 A-Level 考试中，学生经常需要化简包含阶乘的表达式。关键策略是部分展开较大的阶乘，以揭示公因子。

For example, to simplify n! / (n − 2)!:

例如，化简 n! / (n − 2)!：

n!/(n − 2)! = [n × (n − 1) × (n − 2)!]/(n − 2)! = n × (n − 1)

Similarly, (n + 1)! / n! = n + 1. The general principle is to write the numerator in terms of the denominator’s factorial and then cancel.

类似地，(n + 1)! / n! = n + 1。一般原则是将分子写成包含分母阶乘的形式，然后进行约分。

When simplifying, always look for the largest factorial that appears in both numerator and denominator, and express other factorials in terms of it.

化简时，始终寻找分子和分母中共同出现的最大阶乘，并将其余阶乘用该阶乘表示。

4. Operations with Factorials | 阶乘的运算

Factorials follow specific operational rules when combined with addition, subtraction, multiplication, and division.

阶乘在与加、减、乘、除结合时遵循特定的运算规则。

(n + 1)! = (n + 1) × n!

(n + 2)! = (n + 2) × (n + 1) × n!

These expansions allow us to combine terms with different factorials. For example, to simplify (n + 1)! − n!:

这些展开式使我们可以合并包含不同阶乘的项。例如，化简 (n + 1)! − n!：

(n + 1)! − n! = (n + 1) × n! − 1 × n! = n! × [(n + 1) − 1] = n! × n

This technique of factoring out a common factorial is analogous to factoring out a common algebraic factor.

这种提出公因阶乘的技巧类似于代数中提出公因式的做法。

Note that n! + m! for n ≠ m cannot generally be simplified further without expressing both in terms of the smaller factorial.

注意，当 n ≠ m 时，n! + m! 通常无法进一步化简，除非将两者都用较小的阶乘表示。

5. Factorials in Permutations | 阶乘在排列中的应用

In permutations, factorial notation provides a concise way to count arrangements. The number of ways to arrange n distinct objects in a line is n!.

在排列问题中，阶乘记号提供了一种简洁的计数方式。将 n 个不同物体排成一条直线的方法数为 n!。

The number of permutations of r objects chosen from n distinct objects is denoted as ⁿPᵣ and is given by:

从 n 个不同物体中选取 r 个物体进行排列的方法数记作 ⁿPᵣ，其计算公式为：

ⁿPᵣ = n! / (n − r)!

For instance, the number of ways to arrange 3 books from a collection of 5 books is ⁵P₃ = 5! / 2! = 60.

例如，从 5 本书中选取 3 本进行排列的方法数为 ⁵P₃ = 5! / 2! = 60。

When objects are not all distinct, the formula involves division by factorials of repeated counts. If there are n objects with n₁ identical of one type, n₂ identical of another type, etc., the number of distinct arrangements is n! / (n₁! × n₂! × …).

当物体并非全部不同时，公式涉及除以重复数量的阶乘。若有 n 个物体，其中 n₁ 个属于同一种类，n₂ 个属于另一种类，以此类推，则不同排列的方法数为 n! / (n₁! × n₂! × …)。

6. Factorials in Combinations | 阶乘在组合中的应用

Combinations count the number of ways to select objects without regard to order. The number of combinations of r objects from n objects is:

组合问题不考虑顺序地选取物体。从 n 个物体中选取 r 个物体的组合数为：

ⁿCᵣ = n! / [r! × (n − r)!]

The combination formula is central to binomial expansion. The binomial coefficient ⁿCᵣ is also written as (n r) in many textbooks.

组合公式是二项式展开的核心。二项式系数 ⁿCᵣ 在许多教材中也写作 (n r) 的形式。

A useful identity is ⁿCᵣ = ⁿCₙ₋ᵣ, which follows directly from the symmetry of the formula. For example, ¹⁰C₇ = ¹⁰C₃.

一个有用的恒等式是 ⁿCᵣ = ⁿCₙ₋ᵣ，这直接由公式的对称性得出。例如，¹⁰C₇ = ¹⁰C₃。

Pascal’s rule states that ⁿCᵣ + ⁿCᵣ₊₁ = ⁿ⁺¹Cᵣ₊₁, which can be verified using factorial manipulations.

帕斯卡法则指出 ⁿCᵣ + ⁿCᵣ₊₁ = ⁿ⁺¹Cᵣ₊₁，这可以通过阶乘运算加以验证。

7. Double Factorials and Related Notations | 双阶乘及相关记号

Beyond the standard factorial, A-Level students may encounter double factorials. The double factorial of n, denoted n!!, is the product of all integers from n down to 1 that have the same parity as n.

除了标准阶乘，A-Level 学生还可能遇到双阶乘。n 的双阶乘记作 n!!，定义为从 n 开始递减到 1 且与 n 具有相同奇偶性的所有整数之积。

For example, 7!! = 7 × 5 × 3 × 1 = 105, and 8!! = 8 × 6 × 4 × 2 = 384.

例如，7!! = 7 × 5 × 3 × 1 = 105，而 8!! = 8 × 6 × 4 × 2 = 384。

Double factorials appear in certain integrals and series expansions but are less frequently tested than standard factorials in A-Level examinations.

双阶乘出现在某些积分和级数展开中，但在 A-Level 考试中不如标准阶乘那样频繁考查。

Students should be careful not to confuse n!! with (n!)!. The former is a double factorial, while the latter is the factorial of a factorial, which is vastly larger.

学生应注意不要将 n!! 与 (n!)! 混淆。前者是双阶乘，而后者是阶乘的阶乘，其数值要大得多。

8. Factorials in Binomial Expansion | 阶乘在二项式展开中的应用

The binomial theorem states that for any positive integer n:

二项式定理指出，对于任意正整数 n：

(a + b)ⁿ = Σᵣ₌₀ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ

where ⁿCᵣ = n! / [r! × (n − r)!]. This theorem is a direct application of factorial notation.

其中 ⁿCᵣ = n! / [r! × (n − r)!]。该定理是阶乘记号的直接应用。

For instance, the expansion of (1 + x)⁵ is:

例如，(1 + x)⁵ 的展开式为：

(1 + x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵

The coefficients 1, 5, 10, 10, 5, 1 correspond to ⁵C₀, ⁵C₁, ⁵C₂, ⁵C₃, ⁵C₄, ⁵C₅ respectively.

系数 1, 5, 10, 10, 5, 1 分别对应于 ⁵C₀, ⁵C₁, ⁵C₂, ⁵C₃, ⁵C₄, ⁵C₅。

For negative or fractional powers, the generalised binomial theorem requires the use of factorial-like products, often expressed using the binomial coefficient notation with non-integer upper indices.

对于负指数或分数指数，广义二项式定理需要使用类阶乘乘积，通常用具有非整数上标的二项式系数记号表示。

9. Factorials in Probability | 阶乘在概率中的应用

Factorials are essential in probability calculations involving arrangements and selections. Classical probability often requires counting favourable outcomes and total outcomes using permutations and combinations.

阶乘在涉及排列和选择的概率计算中至关重要。经典概率经常需要使用排列和组合来计算有利结果数和总结果数。

For example, if 3 cards are drawn from a standard deck of 52 cards, the number of possible hands is C(52, 3) = 52! / (3! × 49!) = 22,100.

例如，从一副 52 张的标准扑克牌中抽取 3 张，可能的牌型数为 C(52, 3) = 52! / (3! × 49!) = 22,100。

In arrangements of people around a circular table, the number of distinct seating arrangements is (n − 1)!, because rotations are considered identical.

在围绕圆桌安排人员的场景中，不同的座位安排数为 (n − 1)!，因为旋转被视为相同。

Understanding factorial arithmetic is therefore a prerequisite for solving these probability problems accurately.

因此，理解阶乘运算技巧是准确解决这些概率问题的先决条件。

10. Common Mistakes and Pitfalls | 常见错误与陷阱

Students often make predictable errors when working with factorials. One common mistake is assuming n! + m! = (n + m)!, which is completely false. For example, 2! + 3! = 2 + 6 = 8, while (2 + 3)! = 120.

学生在处理阶乘时经常犯一些可预见的错误。一个常见错误是认为 n! + m! = (n + m)!，这是完全错误的。例如，2! + 3! = 2 + 6 = 8，而 (2 + 3)! = 120。

Another misconception is that n! / m! = (n/m)!. In fact, n! / m! = n × (n − 1) × … × (m + 1) when n > m, which is a product, not a factorial.

另一个误解是 n! / m! = (n/m)!。事实上，当 n > m 时，n! / m! = n × (n − 1) × … × (m + 1)，这是一个乘积，而不是阶乘。

A third common error is forgetting that 0! = 1. In combinatorial formulas, substituting n = r gives ⁿCₙ = n! / (n! × 0!) = 1 / 1 = 1, which relies entirely on the convention 0! = 1.

第三个常见错误是忘记 0! = 1。在组合公式中，代入 n = r 得到 ⁿCₙ = n! / (n! × 0!) = 1 / 1 = 1，这完全依赖于 0! = 1 的约定。

Students should also avoid attempting to distribute operations across factorials. Factorials only apply to the number directly preceding the exclamation mark.

学生还应避免试图对阶乘进行分配律运算。阶乘只作用于感叹号前紧邻的数。

11. Worked Examples | 典型例题

Let us work through several typical A-Level problems involving factorial operations.

让我们来解答几道 A-Level 中涉及阶乘运算的典型题目。

Example 1: Simplify (n + 2)! / n!.

例题 1：化简 (n + 2)! / n!。

(n + 2)! / n! = [(n + 2)(n + 1)n!] / n! = (n + 2)(n + 1)

Example 2: Solve the equation n! = 720 for n.

例题 2：解方程 n! = 720。

By computing factorials: 5! = 120, 6! = 720, so n = 6.

依次计算阶乘：5! = 120，6! = 720，因此 n = 6。

Example 3: Express n! / [(n − 2)! × 2!] in its simplest form.

例题 3：将 n! / [(n − 2)! × 2!] 化为最简形式。

n! / [(n − 2)! × 2!] = [n(n − 1)(n − 2)!] / [(n − 2)! × 2] = n(n − 1) / 2

Example 4: Determine which is larger: ⁵P₂ × 3! or ⁶P₃.

例题 4：判断 ⁵P₂ × 3! 与 ⁶P₃ 哪个更大。

⁵P₂ × 3! = (5!/3!) × 6 = 20 × 6 = 120

⁶P₃ = 6!/3! = 120

Therefore, both expressions are equal to 120.

因此，两个表达式都等于 120。

12. Summary and Exam Tips | 总结与考试技巧

Factorial notation is a concise and powerful tool in A-Level Mathematics. Mastery of factorial operations is essential for success in topics such as permutations, combinations, binomial expansion, and probability.

阶乘记号是 A-Level 数学中简洁而强大的工具。掌握阶乘运算对于在排列、组合、二项式展开和概率等主题中取得好成绩至关重要。

Here are some key exam tips:

以下是一些关键的考试技巧：

Always remember 0! = 1 and 1! = 1.
永远记住 0! = 1 以及 1! = 1。
When simplifying, always expand the larger factorial partially to cancel common factors.
化简时，始终部分展开较大的阶乘以约去公因子。
Do not attempt to add or multiply factorials as if they were ordinary numbers.
不要像对待普通数那样对阶乘进行加法和乘法运算。
In combination problems, check whether ⁿCᵣ = ⁿCₙ₋ᵣ can simplify your calculations.
在组合问题中，检查 ⁿCᵣ = ⁿCₙ₋ᵣ 是否能够简化计算。
Practise converting between factorial notation and expanded products fluently.
熟练练习阶乘记号与展开乘积之间的相互转换。

With consistent practice and attention to detail, factorial manipulation becomes straightforward and reliable.

通过持续练习和注意细节，阶乘运算将变得简单而可靠。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Factorial Notation and Its Operational Rules | 阶乘记号及其运算规则