Question Number 2932 by Filup last updated on 30/Nov/15
![What is the subfactorial function? i.e. !x](https://www.tinkutara.com/question/Q2932.png)
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{subfactorial}\:\mathrm{function}? \\ $$$${i}.{e}.\:\:\:\:\:!{x} \\ $$
Commented by 123456 last updated on 30/Nov/15
![number of dessaragment](https://www.tinkutara.com/question/Q2940.png)
$$\mathrm{number}\:\mathrm{of}\:\mathrm{dessaragment} \\ $$
Commented by Rasheed Soomro last updated on 01/Dec/15
![You mean ′disarrangement′?](https://www.tinkutara.com/question/Q2943.png)
$$\mathcal{Y}{ou}\:{mean}\:'{disarrangement}'? \\ $$
Commented by 123456 last updated on 02/Dec/15
![yes](https://www.tinkutara.com/question/Q2988.png)
$$\mathrm{yes} \\ $$
Answered by 123456 last updated on 02/Dec/15
![number of dessaragment !3=2 (1,2,3)⇒(3,1,2),(2,3,1) !n=n∙!(n−1)+(−1)^n !n≤n!](https://www.tinkutara.com/question/Q2989.png)
$$\mathrm{number}\:\mathrm{of}\:\mathrm{dessaragment} \\ $$$$!\mathrm{3}=\mathrm{2} \\ $$$$\left(\mathrm{1},\mathrm{2},\mathrm{3}\right)\Rightarrow\left(\mathrm{3},\mathrm{1},\mathrm{2}\right),\left(\mathrm{2},\mathrm{3},\mathrm{1}\right) \\ $$$$!{n}={n}\centerdot!\left({n}−\mathrm{1}\right)+\left(−\mathrm{1}\right)^{{n}} \\ $$$$!{n}\leqslant{n}! \\ $$