Permutations

The number n! means n factorial, but what does it represent   

\(n! = (n-1)! \cdot n\)

Example:

\(\begin{array}{rcll} 6! &=& (6-1)! \cdot 6 = 5! \cdot 6 \\ 5! &=& (5-1)! \cdot 5 = 4! \cdot 5 \\ 4! &=& (4-1)! \cdot 4 = 3! \cdot 4 \\ 3! &=& (3-1)! \cdot 3 = 2! \cdot 3 \\ 2! &=& (2-1)! \cdot 2 = 1! \cdot 2 \\ 1! &=& (1-1)! \cdot 1 = 0! \cdot 1 \\ 0! &=& 1 \qquad (\text{by definition})\\\\ 6! &=& 5! \cdot 6 \qquad | \qquad 5! = 4! \cdot 5\\ 6! &=& 4! \cdot 5 \cdot 6 \qquad | \qquad 4! = 3! \cdot 4\\ 6! &=& 3! \cdot 4 \cdot 5 \cdot 6 \qquad | \qquad 3! = 2! \cdot 3 \\ 6! &=& 2! \cdot 3 \cdot 4 \cdot 5 \cdot 6 \qquad | \qquad 2! = 1! \cdot 2 \\ 6! &=& 1! \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \qquad | \qquad 1! = 0! \cdot 1 \\ 6! &=& 0! \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \qquad | \qquad 0! = 1 \qquad (\text{by definition})\\ 6! &=& 1 \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \end{array}\)

The number n! means n factorial, but what does it represent  

n! = 1 * 2* 3* ..... *n

so

6! = 1*2*3*4*5*6