How would you go about writing an expression for:
"The number of subsets of $\{1,2,3, . . . ,10\}$ with three elements, that contain at least one odd number and at least one even number."
I can work out an expression for each of these, but I am not sure how to write an expression that combines all three conditions ($3$ elements, at least one odd & even) - there would be less subsets for an expression when conditions are combined.
Appreciate any help.
On
Set $E := \{2,4,6,8,10\}$ the set of even numbers and $O = \{1,3,5,7,9\}$ the set of odd numbers. Then you can describe your sets $M \subseteq \{1,2,\ldots, 10\}$ by the single condition
$$|M \cap E| \cdot |M \cap O| \geq 2.$$
If that is better than having multiple conditions is a different question of course.
You could count all 3-element subsets and then subtract the ones that are all odd or all even. This gives ${10 \choose 3} - {5 \choose 3} - {5 \choose 3}$.