Basic Principle of Enumeration (if a few elements) or Multiplication (if lots of elements)

1a. **Addition Theorem**:

If a work completes in a single level and we have different independent options to do it, then it can be completed by the number of ways equals to the addition theorem of basic principle of enumeration.

1b. **Addition Principle**

If first operation can be performed in *m* ways and another operation, which is independent of the first, can beperformed in *n* ways. Then, either of the two operations can be performed in *m+n* ways. This can be extended to any finite number of exclusife events.

2a. **Multiplication theorem**:

If a work completes in more then one level and we have different independent options to do it, then it can be completed by the number of ways equals to the product of the options called the multiplication theorem of basic principle of enumeration.

2b. **Multiplication Principle**

If first operation can be performed in *m* ways and then a second operation can be performed in *n* ways. Then, the two operations taken together can be performed in *m×n* ways. This can be extended to any finite number of operations.

**Multiplication Principle**:

Consider the 3 letter words that can be made from the letters WORD if no letter is repeated There are:

4 ways of choosing the 1st letter

3 ways of choosing the 2nd letter

2 ways of choosing the 3rd letter

Number of words =4×3×2=24

This is an illustration of the multiplication principle i.e., if several operations are carried out in a certain order, then the number of ways of performing all the operations is the product of the numbers of ways of performing each operation.

If on operation can be performed in *m* different ways and another operation in *n* different ways then these two operations can be performed one after the other in *m×n* ways.

Example 1. Baskin Robbins has 20 flavours of ice cream and 11 flavours of sherbet. In how many ways could you select

a. A scoop of ice cream __or__ a scoop of sherbet?

b. A scoop of ice cream __and__ then a scoop of sherbet?

🔑

a. 20+11=31

b. 20×11=220

**Counting Sample Spaces**

Below we will discuss how to ﬁnd the number of sample points in a sample space or in an event.

Rule 1: **Multiplication**

If an experiment produces *m* results (outcomes) and for each result the second experiment produces *n* results then there is a total of *m×n* results.

Example 2

If two coins are tossed, then

the total number of sample points =2×2=4.

In general, if *n*, coins are tossed, the total sample points is 2^{n}.

Example 3

If two dice are rolled, then

the total number of sample points =6×6=36.

In general, if *n* dice are rolled, the total sample points =6^{n}.

Example 4: **Enumeration**

If a die is rolled and a coin is tossed simultaneously, then the total number of sample points =6×2=12, and the sample space is

*S*={1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}.

Note: H is a Head side and T is a Tail side.

Example 5

In a multiple choice test with 4 questions each of 3 choices, the number of ways a student can answer is 3×3×3×3=81.

🌈 Using tree diagrams to enumerate parallelable occurrence of an event

Rule 2: Permutations

Rule 3: Combinations

Rule 4: Venn Diagram