Probability defines the measure of the occurrence of a likable event out of all the possible results/outcomes of that event. The probability of an event always ranges between 0 to 1. When Probability is zero it signifies that there is no chance for the favorable/likable outcome to occur. And on the other hand, when Probability is one, it signifies that there is a certainty that every time the likable outcome will appear as the outcome of the event. Knowledge of Numbers thus becomes very important when considering Probability scenarios.

The fundamental principle of counting

The Fundamental Probability of Counting suggests that if there is a probability scenario where there are x₁, x₂, x₃ … x_n entity objects each with y₁, y₂, y₃ … y_n choices available for each of the entity then the number of ways,

Ways = y₁ × y₂ × y₃ × … × y_n

Here, the choices for each available entity are multiplied to get the total ways of selection.

What is the importance of the Fundamental Principles of Counting?

Answer:

The Fundamental Principle of Counting is essential as it has the below characteristics:

• Fundamental Principle of Counting  helps to determine how selection is done on the basis of available choices.
• Fundamental Principle of Counting simplifies the approach of selection considering all the possible choices and their combinations for calculating the Probability.
• The Fundamental Principle of Counting is one such vital part of Probability which deals with the knowledge of numbers and there much-needed use when considered from the knowledge of Mathematics.

Example of Fundamental Counting Principle Problem, Consider Seema has 2 blue pens, 2 black and 2 red pens. In how many ways can she select one pen of each kind,

Then pairing can take place as follows:

(B₁ b₁ r₁), (B₁ b₁ r₂), (B₁ b₂ r₁), (B₁ b₂ r₂), (B₂ b₁ r₁), (B₂ b₁ r₂), (B₂ b₂ r₁), (B₂ b₂ r₂)

The total number of ways of choosing this pairing using Counting Principle Problems,

• Choices available for  blue pens = 2
• Choices available for black pens = 2
• Choices available for red pens = 2

Total number of ways: 2 × 2 × 2 = 8

Similar Problems

Question 1: Does Fundamental Counting Principle always hold for Counting Problems.

Answer: 

Yes, Fundamental Counting Principle always holds for Counting Problems.

Question 2: Consider a teacher who has 1 black and 2 red pens. In how many ways can she select one pen of each kind.

Solution: 

Then pairing can take place as follows,

(B₁ R₁), (B₁ R₂)

• Choices available for black pens = 1
• Choices available for red pens = 2

Total number of ways: 1 × 2 = 2

Question 3: Consider a boy has choice of selecting between 2 cups of tea and 2 cups of coffee. In how many ways can he select one cup of each kind.

Solution: 

Then pairing can take place as follows,

(T₁ C₁), (T₁ C₂), (T₂ C₁), (T₂ C₂) 

• Choices available for tea = 2
• Choices available for coffee = 2

Total number of ways: 2 × 2  = 4

Question 4: Consider a child has 3 lollipop and 2 toffees. In how many ways can she select one candy of each kind.

Answer: 

Then pairing can take place as follows,

(L₁ T₁), (L₁ T₂), (L₂ T₁), (L₂ T₂), (L₃ T₁), (L₃ T₂)

• Choices available for black pens = 3
• Choices available for red pens = 2

Total number of ways: 3 × 2 = 6

Question 5: Consider a girl has 3 black tops and 2 blue lowers. In how many ways can she select a dress combo of top and lower.

Solution: 

Then pairing can take place as follows:

(B₁ b₁), (B₁ b₂), (B₂ b₁), (B₂ b₂), (B₃ b₁), (B₃ b₂)

• Choices available for black tops = 3
• Choices available for blue lowers = 2

Total number of ways: 3 × 2 = 6