> ## Documentation Index
> Fetch the complete documentation index at: https://v1-learn.neoartd.my.id/llms.txt
> Use this file to discover all available pages before exploring further.

# Tree Diagrams and the Multiplication Rule for Counting

## Tree Diagrams

A `tree diagram` is a device used to list all possibilities of a sequence of events in a systematic way.

> `diagram pohon` adalah alat yang digunakan untuk mencantumkan semua kemungkinan dari urutan peristiwa secara sistematis.

### Tree Diagrams - Example

<Card>
  Suppose a sales person can travel from New York to Pittsburgh by plane, train, or bus, and from Pittsburgh to Cincinnati by bus, boat, or automobile. Display the information using a tree diagram.

  <br />

  <img src="https://mintcdn.com/mmn/jkRUy8NW28zFUOsL/images/tree-diagrams1.png?fit=max&auto=format&n=jkRUy8NW28zFUOsL&q=85&s=c6a1b5fc2ece8f06f3aee908225a65ce" alt="Tree Diagrams - Example" width="486" height="328" data-path="images/tree-diagrams1.png" />
</Card>

## The Multiplication Rule for Counting

`Multiplication Rule` : In a sequence of **$n$** events in which the first one has **$k_1$** possibilities and the second event has **$k_2$** and the third has **$k_3$**, and so forth, the total possibilities of the sequence will be **$k_1 \times k_2 \times k_3 \times ... \times k_n$**.

### The Multiplication Rule for Counting - Example

<Card>
  A nurse has three patients to visit. How many different ways can she make her rounds if she visits each patient only once?

  <br />

  Solution :

  She can choose from three patients for the first visit and choose from two patients for the second visit, since there are two left. On the third visit, she will see the one patient who is left. Hence, the total number of different possible outcomes is $3 \times 2 \times 1 = 6$.
</Card>

<Card>
  Employees of a large corporation are to be issued special coded identification cards. The card consists of 4 letters of the alphabet. Each letter can be used up to 4 times in the code. How many different ID cards can be issued?

  <br />

  Solution:

  Since 4 letters are to be used, there are 4 spaces to fill **( \_ \_ \_ \_ )**. Since there are 26 different letters to select from and each letter can be used up to 4 times, then the total number of identification cards that can be made is $26 \times 26 \times 26 \times 26 = 456,976$.
</Card>

<Card>
  The digits 0, 1, 2, 3, and 4 are to be used in a 4-digit ID card. How many different
  cards are possible if repetitions are permitted?

  <br />

  Solution:

  Since there are four spaces to fill and five choices for each space, the solution is $5 \times 5 \times 5 \times 5 = 5^4 = 625$.

  <br />

  What if the repetitions were not permitted in the previous example?

  <br />

  Solution:

  The first digit can be chosen in five ways. But the second digit can be chosen in only four ways, since there are only four digits left; etc. Thus the solution is $5 \times 4 \times 3 \times 2 = 120$.
</Card>