Experiments you can run yourself.
The numbers in the nth row of Pascal’s triangle are the coefficients of the binomial (a+b)ⁿ.
The number of outcomes of a sequence of independent choices = the product of the option counts.
Which counting model to use depends on whether order matters and whether repetitions are allowed.
C(n,k)=C(n,n−k): choosing k elements is the same as rejecting n−k.
N objects in M<N boxes ⇒ at least two objects share a box.