Factorial Notation in A-Level Mathematics | A-Level数学：阶乘记法

📚 Factorial Notation in A-Level Mathematics | A-Level数学：阶乘记法

The factorial of a non-negative integer n, written as n!, is one of the most useful notations in A-Level mathematics. It condenses a long product into a single symbol, making counting and algebra far cleaner.

在A-Level数学中，非负整数 n 的阶乘记作 n!，是最实用的记法之一。它把一个长乘积压缩成一个符号，使计数与代数运算变得简洁很多。

1. What Is Factorial Notation? | 什么是阶乘记法？

For a positive integer n, the expression n! means the product of all positive integers from 1 up to n:

对于正整数 n，表达式 n! 表示从 1 到 n 的所有正整数的乘积：

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

For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The symbol ‘!’ is read as ‘factorial’.

例如，5! = 5 × 4 × 3 × 2 × 1 = 120。符号“!”读作“阶乘”。

2. Evaluating Factorials | 计算阶乘

To evaluate a factorial, multiply all integers from n down to 1. The values grow very quickly: 6! = 720, 7! = 5040 and 8! = 40320.

计算阶乘时，把从 n 到 1 的所有整数相乘。阶乘的值增长非常快：6! = 720，7! = 5040，8! = 40320。

In A-Level questions you may be asked to give answers as exact integers or in factorial form. Always read the question carefully.

在A-Level题目中，有时要求写出精确整数，有时要求保留阶乘形式。做题时务必仔细审题。

3. The Recursive Property | 递归性质

A vital property of factorials is the recurrence relation:

阶乘的一个重要性质是递归关系：

n! = n × (n − 1)!

This allows larger factorials to be built from smaller ones. For example, 6! = 6 × 5! = 6 × 120 = 720.

这个关系允许我们用较小的阶乘构造较大的阶乘。例如，6! = 6 × 5! = 6 × 120 = 720。

It is also the key to simplifying expressions such as (n+1)!/n! = n + 1.

它也是化简类似 (n+1)!/n! = n + 1 这类表达式的关键。

4. Why Is 0! Equal to 1? | 为什么 0! 等于 1？

By definition, 0! = 1. This may seem strange, but it is consistent with the recurrence relation: 1! = 1 × 0! gives 1! = 1 × 1 = 1.

根据定义，0! = 1。这看起来可能奇怪，但它与递归关系保持一致：1! = 1 × 0!，即 1! = 1 × 1 = 1。

It also makes formulas work for edge cases. For example, arranging zero objects has exactly one way: do nothing.

这个约定也使公式在边界情况下成立。例如，排列零个物体恰好有一种方式：什么都不做。

5. Factorials and Permutations | 阶乘与排列

Factorials are central to counting arrangements. The number of ways to arrange r objects selected from n distinct objects in order is given by:

阶乘在计数排列中至关重要。从 n 个不同物体中按顺序选出 r 个物体进行排列的方法数为：

P(n, r) = n! / (n − r)!

For instance, the number of ways to choose and arrange 3 letters from the word ‘MATHS’ is P(5, 3) = 5! / 2! = 120 / 2 = 60.

例如，从单词“MATHS”中选出并排列 3 个字母的方法数为 P(5, 3) = 5! / 2! = 120 / 2 = 60。

The special case r = n gives n!, because arranging all n objects in a line has n! possible orders.

当 r = n 时，得到 n!，因为将全部 n 个物体排成一行共有 n! 种顺序。

6. Factorials and Combinations | 阶乘与组合

When order does not matter, we use combinations. The number of ways to choose r objects from n distinct objects is:

当顺序不重要时，我们使用组合。从 n 个不同物体中选取 r 个物体的方法数为：

C(n, r) = n! / (r! × (n − r)!)

For example, choosing 2 representatives from a group of 10 gives C(10, 2) = 10! / (2! × 8!) = (10 × 9) / 2 = 45.

例如，从 10 人中选出 2 名代表，有 C(10, 2) = 10! / (2! × 8!) = (10 × 9) / 2 = 45 种方法。

Notice how the factorial expression simplifies before we multiply: this avoids large numbers and reduces errors.

注意在相乘之前可以化简阶乘表达式：这样可以避免大数运算，减少错误。

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

A common exam skill is simplifying ratios of factorials. Always expand the larger factorial until you reach the smaller one.

常见的考试技能是化简阶乘的比例式。通常应把较大的阶乘展开，直到接触到较小的阶乘。

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

Expanding might be useful, but leaving the answer in factored form is often acceptable in A-Level unless asked otherwise.

展开可能有用，但在A-Level中，除非题目特别要求，保留因式分解形式通常是可以接受的。

Simplify before using a calculator.
Use the recurrence relation when possible.
Check if the question asks for an exact value or a simplified algebraic form.

先化简，再使用计算器。
尽可能使用递归关系。
注意题目要求的是精确数值还是代数化简形式。

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

The binomial theorem uses factorials to express coefficients. For a positive integer n:

二项式定理使用阶乘来表示系数。对于正整数 n：

(a + b)ⁿ = Σ [n! / (r! × (n − r)!)] × a^(n−r) × b^r

The coefficient of each term is exactly C(n, r), also written as nCr or n choose r.

每一项的系数就是 C(n, r)，也写作 nCr 或组合数 n choose r。

For example, the coefficient of x² in (1 + x)⁵ is C(5, 2) = 5! / (2! × 3!) = 10.

例如，(1 + x)⁵ 中 x² 的系数为 C(5, 2) = 5! / (2! × 3!) = 10。

9. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

Students often confuse n! with repeated multiplication nⁿ. Remember that 4! = 24, while 4⁴ = 256.

学生常把 n! 与幂运算 nⁿ 混淆。记住 4! = 24，而 4⁴ = 256。

Another common error is to assume (n + 1)! = n! + 1. This is false; use the recurrence relation instead.

另一个常见错误是认为 (n + 1)! = n! + 1。这是错误的；应当使用递归关系。

Examiners often test whether you can cancel factorials before multiplying. For example, 5! / 3! = 20, not 40/6.

考官常测试你是否能先约分再相乘。例如，5! / 3! = 20，而不是 40/6。

Always check your calculator’s factorial button and know whether it is under MATH or PROBABILITY mode.

要熟悉计算器上的阶乘按键，确认它位于数学还是概率菜单下。

10. Summary and Practice Advice | 总结与练习建议

Factorial notation is a compact and powerful tool in permutations, combinations and binomial expansions. Mastering it early makes later topics much easier.

阶乘记法是排列、组合和二项展开中简洁而强大的工具。尽早掌握它，会让后续内容轻松很多。

Try to rewrite expressions using recurrence and cancellation before calculating. Practice with small numbers first, then move to algebraic forms such as (n + 1)! / (n − 1)!.

练习时先利用递归关系和约分改写表达式，再进行计算。先用小数字练习，再过渡到 (n + 1)! / (n − 1)! 这类代数形式。

Memorise the first few factorials: 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720.

记住前几个阶乘值：0! = 1，1! = 1，2! = 2，3! = 6，4! = 24，5! = 120，6! = 720。

In exams, show your working in factorial form first, then simplify. This makes your method clear and reduces arithmetic mistakes.

考试中，先以阶乘形式写出过程，再化简。这样既能让解题步骤清晰，也能减少计算错误。

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Factorial Notation in A-Level Mathematics | A-Level数学：阶乘记法