In this post we will try to understand the concept of counting principle which is important part of Grade 11 math chapter Permutation & Combination.
For our understanding we will first understand the concept with the help of example and then move on to its technical definition which you have to write in your class 11 cbse math exam
What is Fundamental Counting Principle?
There is a boy named Adam who is getting dressed up for a party.
In his wardrobe, he has following choices: 3 Shirts and 2 Pants.
What are the different options he can wear in the party?
From the above illustration it is clear that:
(a) Adam has three choices of shirts
(b) For every shirt, he has two choices of pants
Therefore total choice of dress combination => 3 (shirts) * 2 (Pants) => 6
This is how fundamental principal of counting works.
Counting Principle Technical Definition
Definition
” If any even occurs in m different ways after that another event occurs in n different ways, then the total occurrence of events can be given as m * n “
In the above example, the first even is the number of shirt which occurs 3 times, following which there is another event i.e. number of pants, which occurs 2 times, then the total combination of event is 3 * 2 = 6
Note: Understand that for this principle, both the events should be independent of each other.
For example in the above case where boy has to choose his dress, the events are independent on each other.
a. Event 01 : Selection of shirt
b. Event 02 : Selection of trouser
In the above case the boy can choose any type of shirt or trouser for the party. Both the decisions are not getting influenced with each other, hence are independent.
Example 02
Suppose you decided to buy a new car and you have shortlisted three options “Honda”, “Suzuki” and “Toyota”. All the cars have same color options ” Red, Yellow, Blue “ and with different seating options “Two seater and Five Seater”
Find the number of choices you have?
Solution
Brand : Honda, Suzuki, Toyota
Colors : Red, Yellow, Blue
Seating : Two, Five
You can observe that all the choices are independent on each other, so you can apply above mentioned multiplication principle for this question.
Using the concept of fundamental counting, the number of options of different car is
==> 3 * 2 *3
==> 18
We have total of 18 options of car to buy.
Events are non independent
I have already said that the multiplication principle of counting has its limitation, It only works when the events provided are independent of each other.
But what if in a scenario, the events are non independent. In this case you have to solve question manually without the help of any formula.
Example
Suppose in the case discussed above, it has been found that
1. “Yellow” color is not available in “Honda” cars
2. “Suzuki” cars do not have “5 Seat” option
You can see that some of the color and seating choices is dependent on the car brand, so you cannot apply that multiplication formula for this question.
To solve this case, we will draw graphical illustration of the choices of car available and will try to count the number of option available to us.
In this case we have to manually count all the available options to find the right answer.
For this question, the total number of option available is 13