Factorial

Definitions

The 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.

Equivalents

Examples

“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⟳!”

CEFR level