If we attempt to write $(1)$ as a product of odd transpositions, say $(1)=\underbrace{(a_1 a_2)(a_3 a_4)...(a_m a_n)}_{k\text{ times}}$, where $k$ is odd, then what is the property of a product of an even number of transpositions, such that we arrive at the contradiction that $(1)$ cannot be written as a product of k transpositions?
I still don't have a lot of intuition for symmetric groups.