Counting

Multiplication Rule

Let’s say we have a compound experiment (an experiment with multiple components). If the 1st component has \(n_1\) possible outcomes, the 2nd component has \(n_2\) possible outcomes, \(\dots\), and the \(r\)-th component has \(n_r\) possible outcomes, then overall there are \(n_1n_2 \dots n_r\) possibilities for the whole experiment.

Sampling Table

The sampling table gives the number of possible samples of size \(k\) out of a population of size \(n\), under various assumptions about how the sample is collected.

	Order Matters	Not Matters
With Replacement	\(\displaystyle n^k\)	\(\displaystyle{n+k-1 \choose k}\)
Without Replacement	\(\displaystyle\frac{n!}{(n - k)!}\)	\(\displaystyle{n \choose k}\)