# Ch2: Combinatorial Probability

## Key concepts

Definitions

- Permutation: arrangement in some order
- Ordered versus unordered samples: In ordered samples, the order of the elements in the sample matters; e.g., digits in a phone number, or the letters in a word. In unordered samples the order of the elements is irrelevant; e.g., elements in a subset, or lottery numbers.
- Samples with replacement versus samples without replacement: In the first case, repetition of the same element is allowed (e.g., numbers in a license plate); in the second, repetition not allowed (as in a lottery drawing—once a number has been drawn, it cannot be drawn again).

Formula's

- Number of
**permutations**of n objects: n! - Number of
**ordered**samples of size r,**with**replacement, from n objects: - Number of
**ordered**samples of size r,**without**replacement, from n objects: - Number of
**unordered**samples of size r,**without**replacement, from a set of n objects (= number of subsets of size r from a set of n elements) (**combinations**): - Number of
**subsets**of a set of n elements:

## Binomial coefficients

Definition

For n = 1, 2, . . . and k = 0, 1, . . . , n,

Combinatorial Interpretations

- the number of ways to select k objects out of n given objects (in the sense of unordered samples without replacement);
- the number of k-element subsets of an n-element set;
- the number of n-letter HT sequences with exactly k H’s and n − k T’s.

## Rules and properties

The product rule

If an experiment consists of performing k partial experiments and the

The permutation rule

The number of orders or permutations (variations) in which k different things can be arranged is

Properties

When **mutually exclusive** events, then:

Hypergeometric formula

If we draw n times, at random and without replacement, from a set of