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In maths, the factorial of a whole number is what you get when you multiply that number by every whole number below it, down to 1. People write it with a tall bang: 5! means 5 × 4 × 3 × 2 × 1, which is 120.
In maths在數學裡，the factorial of a whole number一個整數的factorial（階乘） is what you get就是把那個數字 when you multiply that number by every whole number below it乘上比它小的每一個整數，down to 1一路乘到1所得到的結果。People write it人們會寫成 with a tall bang加驚嘆號的形式：5!5! means 5 × 4 × 3 × 2 × 1意思是5 × 4 × 3 × 2 × 1，which is 120也就是120。
You meet this when you count how many ways you can line things up. It is a classroom and textbook word, not a street word.
You meet this你會 when you count how many ways you can line things up在計算東西有幾種排列方式時用到這個概念。It is a classroom and textbook word它是課堂和教科書用語，not a street word不是日常口語會用的字。
Though actually, 0! is a special case: it is defined as 1, which surprises a lot of students. The idea is simple, but the numbers get huge very fast — 10! is already 3,628,800.
Though actually不過其實，0! is a special case0!是個特殊情況：it is defined as 1它被定義為1，which surprises a lot of students這常讓很多學生感到意外。The idea is simple概念很簡單，but the numbers get huge very fast但數字增加得非常快——10! is already 3,628,80010!就已經是3,628,800了。
There is also an adjective factorial meaning “about a factor”, but the sense you need here is this multiplying one.
There is also an adjective factorial另外還有一個形容詞factorial meaning "about a factor"意思是「跟因數有關的」，but the sense you need here is this multiplying one但你這裡需要的是這個「連乘」的意思。
組成factor + -ial. From factor + -ial.
同家族的字
- factorin maths, a number you multiply; this word is built from it
原始詞典資料4 筆記錄
以下是外部詞典記錄的用法，原文照列，我們還沒有寫成教學內容。上面的說明才是我們自己寫的；有些用法可能上面已經講過了。
factorialnoun/fækˈtɔːɹi.əl/
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combinatoricsmathematicssciencesThe result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.
- The expression #123;#125;#95;nC#95;r means the number of combinations of n things taken r at a time. It is also written #92;binomnr, and is equal to#92;frac#123;n(n-1)(n-2)(n-3)#92;cdots#92;cdots(n-r#43;1)#125;#123;r#33;#125;or n#33;#47;r#33;(n-r)#33;, in which r#33; is read "factorial r" and denotes the product of all the integral numbers from 1 to r inclusive.
- This playing around with consecutive integers reminded one of the students of factorials, and he asked about products of integers, which I said could be expressed similarly as#92;prod#95;#123;k#61;1#125;ⁿk#61;1#92;cdot 2#92;cdot 3#92;cdotsn.But what does this equal? Well, of course we can say that it is equal to n factorial and write#92;prod#95;#123;k#61;1#125;ⁿk#61;n#33; and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!
factorialadj/fækˈtɔːɹi.əl/
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mathematicssciencesOf or pertaining to a factor or factorial.
- I shall denote, as usual, the factorial product of the numbers up to n by n#33;, but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently definen#33;#33;#61;n#92;cdot(n-2)#33;#33;.Starting with1#33;#33;#61;1,#92;quad 2#33;#33;#61;2,it follows that, for positive values of n,n#33;#33;#61;n#92;cdotn-2#92;cdotn-4#92;cdots,where the last factor is either 1 or 2, according as n is odd or even.
- The definition of addition is an example of definition by induction, sometimes called recursive definition. As another example of this, consider the following inductive definition of the factorial function. We define n#33; for n#92;in#92;mathbbN by the following two properties, (i) 1#33; is defined to be 1, (ii) for all k#92;in#92;mathbbN, we define (k#43;1)#33; to be (k#43;1)#92;timesk#33;.
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Of or pertaining to a factor, a kind of business agent.
- The latter sold the goods to a customer who was cashier to certain creditors of the agents without disclosing the factorial capacity in which they acted.
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datedOf or pertaining to a factory.
資料來源：英文維基詞典 English Wiktionary（CC BY-SA 4.0）。 原文照列，未經改寫。