# What are Counting Principle problems?

Probability is a branch of mathematics that is used to calculate the numerical descriptions of how likely an event is going to happen or we can say that probability deals with the occurrence of a random event. The probability of an event is always taking place between zero and one. It can be expressed in numbers or percentages. In the probability, to find the likelihood of a particular event we must first find the total number of a potential outcome.

**Counting Principle Problems**

Counting Principle Problems are based on the simple fundamental approach of counting, keeping in mind the available options to choose suitable items for a particular selection. Counting Principle problems are best solved using a tree-based structure approach. The available choices are represented as branches of tree. Tree representation also simplifies the understanding of Counting Principle Problems. Counting problems can be best described as:

If there are ‘n’ entities and each of the n entities has m1, m2, m3………………mn options to choose from. Say 1st entity has m1 choices, 2nd entity has m2 choices, 3rd entity has m3 choices and so on.

Then the total ways of the selection of entities would be :

Number of ways for Counting Principle Problems:

m1 x m2 x m3 x m4………………………………..x mn

The best way to learn counting principle problems is through an example:

Example: Consider Vaibhav has 3 mangoes, 3 papaya and 3 apples. In how many ways can he put fruit of one kind in a fruit basket.

Solution:Then pairing can take place as follows:

(M1 P1 A1), (M1 P1 A2), (M1 P1 A3), (M1 P2 A1), (M1 P2 A2), (M1 P2 A3), (M1 P3 A1), (M1 P3 A2), (M1 P3 A3)

(M2 P1 A1), (M2 P1 A2), (M2 P1 A3), (M2 P2 A1), (M2 P2 A2), (M2 P2 A3), (M3 P3 A1), (M3 P3 A2), (M3 P3 A3)

(M3 P1 A1), (M3 P1 A2), (M3 P1 A3), (M3 P2 A1), (M3 P2 A2), (M3 P2 A3), (M3 P3 A1), (M3 P3 A2), (M3 P3 A3)

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

Choices available for mangoes (m) = 3

Choices available for papaya (n) = 3

Choices available for apples (n) = 3

Total no. of ways: 3 X 3 X 3 = 27

### Similar Problems

**Question 1. Consider 3 boys and 3 girls want to team up as pair for a Salsa Dance Competition.**

**Solution:**

Then pairing can take place as follows:

(B1 G1) (B1 G2)

(B2 G1) (B2 G2)

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

Choices available for boys (m) = 2

Choices available for girls (n) = 2

Total no. of ways: 2 x 2 = 4

**Question 2. Consider a boy has three choices of shirt to choose for a party.**

**Solution:**

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

There is one boy and three choices of shirt available.

Total no. of ways: 1 x 3 = 3

**Question 3. Consider there are three boys and there are three choices of shirt to choose for a party.**

**Solution:**

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

There are three boys and three choices of shirt available.

(B1 S1) (B1 S2) (B1 S3)

(B2 S1) (B2 S2) (B2 S3)

(B3 S1) (B3 S2) (B3 S3)

Total no. of ways: 3 x 3 = 9

**Question 4. Consider Seema to choose from 4 choices available for vegetables and 2 choices available for breads. In how many ways can she combine vegetable and breads to prepare dinner?**

**Solution:**

Choices available for vegetables: 4

Choices available for bread: 2

(V1 B1) (V1 B2)

(V2 B1) (V2 B2)

(V3 B1) (V3 B2)

(V4 B1) (V4 B2)

Total no. of ways: 4 x 2 = 8

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