# What is the importance of the Fundamental Principles of Counting?

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 x_{1}, x_{2}, x_{3 }… x_{n} entity objects each with y_{1}, y_{2}, y_{3 }… y_{n} choices available for each of the entity then the number of ways,

Ways = y_{1} × y_{2} × y_{3} × … × 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:

(B

_{1}b_{1}r_{1}), (B_{1}b_{1}r_{2}), (B_{1}b_{2}r_{1}), (B_{1}b_{2}r_{2}), (B_{2}b_{1}r_{1}), (B_{2}b_{1}r_{2}), (B_{2 }b_{2}r_{1}), (B_{2}b_{2}r_{2})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,

(B_{1}R_{1}), (B_{1}R_{2})

Choices available for black pens = 1Choices 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,

(T_{1}C_{1}), (T_{1}C_{2}), (T_{2}C_{1), (T2 C2) }

Choices available for tea = 2Choices 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,

(L_{1}T_{1}), (L_{1}T_{2}), (L_{2}T_{1}), (L_{2}T_{2}), (L_{3}T_{1}), (L_{3}T_{2})

Choices available for black pens = 3Choices 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:

(B_{1}b_{1}), (B_{1}b_{2}), (B_{2 }b_{1}), (B_{2 }b_{2}), (B_{3 }b_{1}), (B_{3}b_{2})

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

Total number of ways: 3 × 2 = 6