# How many of the following permutations are even?

This is from a problem from a past exam.

How many of the following permutations on the set $$\{1,2,3,4,5,6,7,8\}$$ are even?

$$(a) \quad \left(\begin{array}{llllllll}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ {2} & {1} & {4} & {3} & {6} & {7} & {8} & {5}\end{array}\right)$$

$$(b) \quad \left(\begin{array}{llllllll}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ {1} & {4} & {3} & {8} & {5} & {6} & {7} & {2}\end{array}\right)$$

$$(c)\quad \left(\begin{array}{llllllll}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ {1} & {2} & {5} & {4} & {7} & {6} & {3} & {8}\end{array}\right)$$

$$(d) \quad (12)(78)$$

$$(e) \quad (345)(678)$$

$$(f) \quad (1234)(5678)$$

All of the are even except $$(a)$$ and $$(b)$$. According to the instructor there should be 5, but I only found 4 of them to be even . I think he has made a mistake.

Note that $$\quad \left(\begin{array}{llllllll}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ {1} & {4} & {3} & {8} & {5} & {6} & {7} & {2}\end{array}\right)$$

is also even.
It is the same as $$(2,4,8) = (2,4)(2,8)$$