🔢 乘法原理计算器（基本计数原理）Fundamental Counting Principle Calculator
输入每一步的选择数（如 3 5 2），按乘法原理求总方案数并列出乘法过程；也可以按“每步选择数相同、共 k 步”计算 nᵏ（例如密码或车牌组合）。使用大整数，结果很大时也能给出精确值和位数。本地处理。Enter the number of choices at each stage (such as 3 5 2) to get the total number of outcomes by the multiplication principle with the working shown, or use the same n choices over k stages for nᵏ (such as PINs or licence plates). Big integers give the exact value and number of digits even for huge results. Processed locally.
使用说明与常见问题How-to & FAQ
「乘法原理计算器（基本计数原理）」是什么？What is the Fundamental Counting Principle Calculator tool?
输入每一步的选择数（如 3 5 2），按乘法原理求总方案数并列出乘法过程；也可以按“每步选择数相同、共 k 步”计算 nᵏ（例如密码或车牌组合）。使用大整数，结果很大时也能给出精确值和位数。本地处理。Enter the number of choices at each stage (such as 3 5 2) to get the total number of outcomes by the multiplication principle with the working shown, or use the same n choices over k stages for nᵏ (such as PINs or licence plates). Big integers give the exact value and number of digits even for huge results. Processed locally.
如何使用「乘法原理计算器（基本计数原理）」？How do I use the Fundamental Counting Principle Calculator tool?
在「乘法原理计算器（基本计数原理）」中，填写「方式」、「各步选择数（空格或逗号分隔）」、「每步选择数 n」、「步数 k」等参数，结果会随输入实时计算并显示在下方。In the Fundamental Counting Principle Calculator: Fill in "Method", "Choices per stage (space or comma separated)", "Choices per stage n", "Stages k"; results are recalculated instantly as you type.
乘法原理是什么？What is the multiplication principle?
如果一件事分成几个相互独立的步骤，第一步有 m 种做法、第二步有 n 种……那么总共有 m × n × … 种做法。例如 3 件上衣、5 条裤子、2 双鞋可以搭配 30 种。If a task has independent stages with m ways for the first, n for the second and so on, there are m × n × … ways in total. 3 tops, 5 trousers and 2 pairs of shoes make 30 outfits.
和加法原理有什么区别？How does it differ from the addition principle?
乘法原理用于“分步完成”（每步都要做），加法原理用于“分类完成”（只选其中一类）。Multiply when every stage must be done in turn; add when you choose just one of several separate categories.
4 位数字密码有多少种？How many 4-digit PINs are there?
每位 0–9 共 10 种，4 位就是 10⁴ = 10,000 种。选择“每步相同”，n = 10、k = 4 即可。Each digit has 10 options, so 10⁴ = 10,000. Choose same choices with n = 10 and k = 4.
能举个计算实例吗？Can you show a worked example?
例如在「乘法原理计算器（基本计数原理）」中填 方式 = 每步选择数，各步选择数（空格或逗号分隔） = 3 5 2，结果区显示：总方案数：30；阶段数：3；计算：3 × 5 × 2 = 30For example, entering Method = Choices per stage, Choices per stage (space or comma separated) = 3 5 2 in the Fundamental Counting Principle Calculator gives: Total outcomes: 30; Stages: 3; Working: 3 × 5 × 2 = 30
哪些输入会被检查？Which inputs are validated?
「乘法原理计算器（基本计数原理）」会检查这些输入——「每步选择数 n」、「步数 k」：所选方式用到时必须填写；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Fundamental Counting Principle Calculator checks its inputs — "Choices per stage n", "Stages k": required when the chosen option uses it; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「乘法原理计算器（基本计数原理）」的设置：「方式」可选 每步选择数、每步相同 nᵏ（默认 每步选择数）；「各步选择数（空格或逗号分隔）」默认 3 5 2；「每步选择数 n」默认 10；「步数 k」默认 4。每次生成结果都按当前设置处理。Settings in the Fundamental Counting Principle Calculator — "Method": Choices per stage, Same choices nᵏ (default Choices per stage); "Choices per stage (space or comma separated)" default 3 5 2; "Choices per stage n" default 10; "Stages k" default 4. The result always uses the current settings.
计算精度如何？How precise are the results?
「乘法原理计算器（基本计数原理）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字），部分功能使用 BigInt 处理大整数。The Fundamental Counting Principle Calculator uses double-precision floating point (about 15–16 significant digits); some features use BigInt for large integers.
「乘法原理计算器（基本计数原理）」免费吗？需要注册吗？Is the Fundamental Counting Principle Calculator free? Do I need an account?
「乘法原理计算器（基本计数原理）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Fundamental Counting Principle Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.