Number Theory


New Expression of the Factorial of $n$ ($n!$, $n\in N$)

Authors: S Maiti

New Expression of the factorial of $n$ ($n!$, $n\in N$) is given in this article. The general expression of it has been proved with help of the Principle of Mathematical Induction. It is found in the form \begin{equation} 1+\sum_{i=1}^{n}a_i +\sum_{\substack{i,j=1 \\(i<j)}}^{n}a_ia_j + \sum_{\substack{i,j,k=1 \\(i<j<k)}}^{n}a_ia_ja_k +\cdots +a_1a_2\cdots a_{n}, \label{factorial_expression} \end{equation} where $a_i=i-1$ for $i=1,~2,~\cdots ,~n$. More convenient expression of this form is provided in Appendix.

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[v1] 2012-07-23 15:12:20

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