Alternating even and odd digits

Text

In how many ways can the six digits $1, 2, 3, 4, 5, 6$ (each used once) be arranged in a row so that even and odd digits alternate?

Solution

There are $3$ odd digits ($1,3,5$) and $3$ even ($2,4,6$). Alternation is possible in two patterns: $O\,E\,O\,E\,O\,E$ or $E\,O\,E\,O\,E\,O$. In each pattern the $3$ odd digits fill their $3$ reserved positions in $3!$ ways and the $3$ even ones fill the others in $3!$ ways. With the two patterns: $N = 2\cdot 3!\cdot 3! = 2\cdot 6\cdot 6 = \boxed{72}.$