I'm going to teach very elementary combinatorics (limited to basic enumeration) during two weeks to middle school students. At the beginning, I want to demonstrate the importance of counting in real life or technology using concrete examples which students are familiar with, so that they appreciate what they are learning.
In this discussion, almost all examples are advanced and are not appropriate for middle school students.
I'm looking forward to your suggestions. Thanks.
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A really simple application of counting principles is to figure out how many digits you need in a telephone number to service a certain sized popuation or how many letters/numbers you need on a license plate in a given state. A nice example of the latter is to bring in slides with license plates from each state. For example, Rhode Island uses 2 letters and 3 numbers, while California uses 3 letters and 4 numbers. See if they can find the state whose license plate design is most efficient/inefficient.
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State lotteries.
(Why, oh beautiful pristine subject, must you go roll in the mud with the gamblers? Sigh...)
;-)
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for openers, I think it is nice to ask your students to list all permutations of the 36 symbols 0..9 A..Z ; then guesstimate that, at one silicon atom per bit, how many meters thickness of the entire earth's surface such list would take (using Stirling's approx. to 36!)
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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.
Calculating the odds of having the winning hand in a poker game.