How many $3$-subsets of $\{1,2,\ldots,10\}$ contain at least one even and one odd integer?

Find the number of three element subsets $A$ of $\{1,2,...,10\}$ such that $A$ contains at least one even and one odd integer.

I think in this way: First we distribute one even and one odd number to 3 sets.Then there are 4 digits left of which 2 are even and 2 are odd.But how to distribute this 4 digits into 3 sets?

I think you might be able to get what you want by determining how many subsets of three elements you can get (with no constraints), and remove all subsets which only have even or odd integers in it.

Which would be: $$\binom{10}{3}-2\binom{5}{3}=100$$

This seems to be where you're heading:

• Step 0: We start with an empty set.

• Step 1: We add in an odd number; which can be done in $5$ ways.

• Step 2: We add in an even number; which can be done in $5$ ways regardless of the choice in Step 1.

• Step 3: We add in any other number; which can be done in $8$ ways regardless of the choices in Steps 1 and 2.

This process generates $5 \times 5 \times 8=200$ (not necessarily distinct) sets. In fact, each set is generated twice. E.g. $\{1,2,3\}$ is generated once when $1$ is picked in Step $1$, and once when $3$ is picked in Step 1. So, we divide the result by $2$.