Parity of three pairwise products from 1 to 6

The numbers $1,\dots,6$ are put in a uniformly random order. Multiply the 1st and 2nd numbers, the 3rd and 4th, and the 5th and 6th, then add the three products. What is the probability that this total is odd?

$3/5$. A product is odd only if both factors are odd, and with three odd numbers at most one pair can be odd-odd. The total is odd exactly when one pair is odd-odd, which happens unless every pair is mixed; the all-mixed probability is $\tfrac35\cdot\tfrac23=\tfrac25$, so the answer is $\tfrac35$.