# 8: Combinatorics

- Page ID
- 8434

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

- 8.1: What is Combinatorics?
- Combinatorics studies the arrangements of objects according to some rules.

- 8.2: Addition and Multiplication Principles
- Use the addition principle if we can break down the problems into cases, and count how many items or choices we have in each case. The total number is the sum of these individual counts. The idea is, instead of counting a large set, we divide it up into several smaller subsets, and count the size of each of them.

- 8.3: Permutations
- An r -permutation of A is an ordered selection of r distinct elements from A . In other words, it is the linear arrangement of r distinct objects. It appears in many other forms and names. The number of permutations of n objects, taken r at a time without replacement. The number of ways to arrange n objects (in a sequence), taken r at a time without replacement.

- 8.4: Combinations
- In many counting problems, the order of arrangement or selection does not matter. In essence, we are selecting or forming subsets.

- 8.5: The Binomial Theorem
- A binomial is a polynomial with exactly two terms. The binomial theorem gives a formula for expanding (x+y)ⁿ for any positive integer n .