Well, knowing that $2n \choose n$ tends to be massive has kept me from trying to write and run a lot of programs that aren't expected to finish before the universe dies of old age, and the point of combinatorics a lot of times is that if it's too big to go through the possibilities you probably don't know how to calculate it properly.
There's the Indian formulation of the Fibonacci sequence and $n \choose k$ as the solution to problems of counting possibilities for poetic meter, unified through Pascal's triangle.
In a similar vein, you can show some campanology patterns and rules and describe how they describe cycling through combinations and permutations.
Probability is pretty fun when you know how to count properly, and you could explain or show how one can manage to make good answers on vast ranges of possibilities with a little application of logic and combinatorics, hence, that by analyzing the situation you can still think about things where the possibilities are too great to be certain just from weighing things out.
Combinatorics problems are a source for a lot of puzzles and games, like the Conway puzzle or Sim, and verbal puzzles like the orchard planting problem or the schoolgirl problem that have geometric importance. What would be of interest is how different the games can be from one another, yet that they all come from the one field.
Even if you couldn't explain to them why, it's worth mentioning that physicists use the number of partitions of a number to describe symmetries in the laws of physics.