Counting rule in statistics

What is counting rule?

Suppose you have some events. The number of the events can be one or more. To find all possible outcomes for these sequence of events counting rule is used. In short, you can say that to find all the possible outcomes from some sequence of events you will use the counting rule

There are three types of counting rule:
1.Fundamental counting rule
2.Permutation rule
3.Combination rule

Fundamental counting rule

Suppose you have a sequence of n events and first event has k₁ possibilities then second event has k₂ and so on. Now the total number of possibilities of the sequence will be:
k₁*k₂* k₃*k₄*...k_n

Example 1:
Pick two letters, suppose C and D, and roll a die. Find the total number of outcomes for the sequence of events.

Here you have two events. First, one is to pick letters and the value of this event is 2 because you have to choose two letters. The second event is rolling a die and the value of this event is 6 because a die has 6 sides. So 6 outcomes will get if you roll a dye.

So,
k₁=2
k₂=6
According to the formula there are 2*6=12 possibilities.

Example 2:
There are 3 major routes from Florida to New York and 5 major routes from New York to Texas. How many different trips can be made from Florida to Texas?
k₁=3
k₂=5
So according to the formula 3*5=15 trips can be made

Permutation rule

In Permutation, you will arrange the events in a specific order. It means here order matters a lot. Permutation use factorial notation. Factorial is all the positive numbers from 1 to n.

Let's see how to calculate factorial:
5!=5*4*3*2*1=120
4!=4*3*2*1=24
8!=8*7*6*5*4*3*2*1=40320
10!=10*9*8*7*6*5*4*3*2*1=3628800

Example 1:
A car buyer has a choice of 10 cars to buy. He decides to rank each car according to certain criteria such as price, specification.
1. How many different way buyers can rank 10 cars?
2.How many different way buyers can rank the top 5 cars?

As there are 10 choice.
So,
10!=10*9*8*7*6*5*4*3*2*1=3628800
So that person can rank in 3628800 ways.

2.For top 5 the answer is 10*9*8*7*6=30240 different ways.
To calculate the top 5 you took the top five numbers from the left side of 10 factorial. If you want to calculate the top 3 then take 3 numbers from the left side of 10 factorial.

1.Here,
n=10 and r=10 because the total number of the event is 10 and you want to output from 10 events.
So, ₁₀P₁₀=10!/(10-10)!=10!/0!=3628800
Note: 0!=1

2.Here,
n=10 and r=5 because the total number of the event is 10 and you want 5 output from 10 event.
₁₀P₅=10!/(10-5)!=10!/5!=30240

Permutation rule 2

Formula: n!/(r₁!*r₂!*r₃!*...r_p!)

Suppose you have a word ACCOUNTING. Now you have to find how many new words you can create by shuffling the words. Like ACCNOUITGN,CACNUOITGN, etc.

Here you have letters A=1,C=2,O=1,U=1,N=2,T=1,I=1,G=1 times and the total number of letters are 10 .
So according to the formula:10!/(1!*2!*1!*1!*2!*1!*1!*1!)=907200
You can say that 907200 words or permutation can be made.

Combination rule

Here you don't need any order means order selection is not important. You use this rule when the order or arrangement is not important in the selection process.

Suppose 20 students are selected from 50 students for research. Here 20 students represent a combination. It doesn't matter who is first and who is second and so on.

Formula: _nC_r=n!/{(n-r)!*r!}
Here,
n=number of events
r= number of events from total number of event

Example:
Suppose 3 students are selected from 5 students for a research. How many combination can possible.
Here,
n=5
r=2
So,
₅C₃=5!/{(5-3)!*3!}=10