Combinatorial counting methods
Combinatorics deals with the number of possible arrangements in a trial.
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Hint
In experiments, which are too extensive for a tree diagram you can also use combinatorial counting methods.
The number of possible outcomes of a k-step experiment is the product of the possible results $n_1$ to $n_k$ of the individual steps.
$n_1\cdot n_2\cdot...\cdot n_k$
Example
A license plate consists of two letters, followed by three digits. How many license plates are possible in total?
- B: Letter (26 options each: A-Z)
Z: Number (10 options each: 0-9) - Structure: $\text{BBZZZ}$
Combinations: $26\cdot26\cdot10\cdot10\cdot10$ $=676.000$
Urn model
Many problems from combinatorics can be traced back to the urn model.
It is a thinking aid in which balls are drawn from an urn. It is important to note whether replacement is allowed or not and whether order matters or not.
The following three models are covered in this article:
- With replacement, order matters (permutation)
- Without replacement, order matters (permutation)
- Without replacement, order doesn't matter (binomial coefficient)