## Probability for machine learning and data science – Basic Probability 3 of 6

In this post we will look at Counting. This post is organised as follows

1. Basic rule of counting
2. Permutations
3. Combinations

### Basic rule of counting

Take k tasks such that task I has n_i ways of doing it, then the total number of ways of doing k tasks is

n_1 \times n_2 \times n_3 \times .... \times n_4

We will see this by using an example

Lets take my favourite biryani restaurant which sells the following

• 3 types of biryani
• 15 types of gravies
• 3 types of desserts

Now that I like biryani I need to choose 1 biryani, 1 gravy and 1 dessert. In how many ways can I do it?

let us break the problem in notation and do it. First we need to determine the k value

The k value for the problem is 3

k = 3

n_1 = 3, n_2 = 15, n_3 = 3

So when using the formula above we have

3 \times 15 \times 3 = 135

So i can choose between 135 possible ways of my food.

Before we wrote

n_1 \times n_2 \times n_3 \times …. \times n_4

we can write this in product notation like this

\prod\limits_{i=1}^{k}n_i

### Permutations

A permutation is defined as the number of ways of ordering n distant objects taken r at a time

^n p_r = n(n-1)(n-2)...(n-r+1)

Factorials

Lets us revise what a factorial is

the notation n! called n factorial is defined as

n! = n(n-1)(n-2)... 2 \times 1

Factorial in product notation

\prod\limits_{i=0}^{n-1}(n-i)

Lets write the permutation formula again

^n p_r = n(n-1)(n-2)...(n-r+1) = \frac{n!}{(n-r)!}

### Combinations

The combination is dealing with when we take objects in which order does not matter.

Combinations can be calculated by using

\binom{n}{r} = \frac{n!}{r!(n-r)!}

You can read more about Counting, Permutations and Combinations from this book.

The topics covered in my posts are from this book.

Happy Learning

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