Fundamental Principle of Counting

Fundamental principle of counting

The fundamental principle of counting is a rule used to find the total number of outcomes possible in a given situation. The fundamental principle of counting can be classified into two types

1. Multiplication Rule (AND rule)
2. Addition Rule (OR rule)

Multiplication Rule: Definition

The multiplication Rule states that for “n” mutually independent events, P₁, P₂, P₃, …P_n. The number of in which these events can occur is n(P₁), n(P₂), n(P₃),… n(P_n) respectively. Now we define an event E such that it is happening all the events simultaneously then the number of ways this can happen is, n( E) = n(P₁) x n(P₂).........................n(P_n)

Formula for Multiplication Rule

According to the multiplication rule, if a certain work W can be completed by doing 2 tasks, first doing task A AND then doing task B. A can be done in m ways and following that B can be done in n ways, then the number of ways of doing the work W is (m x n) ways.

For example, let's say a person wants to travel from Noida to Gurgaon, and he has to travel via New Delhi. It is given that the person can travel from Noida to New Delhi in 3 different ways and from New Delhi to Gurgaon in 5 different ways.

So, in this case, to complete his work (reach Gurgaon) he has to do two tasks one after the other, first traveling from Noida to New Delhi (task A) and then from New Delhi to Gurgaon (task B), as he has 3 different ways of reaching New Delhi (doing task A), and he has 5 different ways to reach Gurgaon from New Delhi (doing task B), so in that way, he has a total of 3×5 = 15 different ways to reach Gurgaon from Noida.

Proof of multiplication rule of the fundamental principle of counting

The first operation can be performed in any one of the m ways and for each of these ways of performing the first operation, there are n ways of performing the second operation.

Thus, if the first operation could be performed in one such way, there would have been 1 x n = n ways of performing both operations. But it is given that the first operation can be performed in m ways and for each way of performing the first operation, the second can be performed in n ways.

Therefore, the total number of ways of performing both operations is n + n + n +… to m terms = n x m.

Note: If three operations can be separately performed in m, n, and p ways, respectively, then the three operations together can be performed in mx n x p ways.

Addition Rule: Definition

The addition Rule states that for two possible events A and B where A and B both are mutually exclusive events, i.e. they have no outcome in common, and if event E is defined as occurring in either event A or event B then the possible number of ways in which event E can occur is n(E) = n(A) + n (B)

Formula for Addition Rule

According to the addition rule, if work W can be completed by doing task A OR task B, and A can be done in m ways and B can be done in n ways (and both cannot occur simultaneously: in this case, we call tasks A and B as mutually exclusive), then work W can be done in (m + n) ways.

Suppose, there are 5 doors in a room: 2 on one side and 3 on the other. A man has to go out of the room. The man can go out from any one of the 5 doors. Thus, the number of ways in which the man can go out is 5. Here, the work of going out through the doors on one side will be done in 2 ways and the work of going out through the doors on the other side will be done in 3 ways. The work of going out will be done when the man goes out from either side I or side II. Thus, the work of going out can be done in 2 + 3 = 5 ways.

For example, let’s say that a person can travel from New Delhi to Noida in 3 different types of buses, and 2 different types of trains, so, he can complete the work of going from New Delhi to Noida in 3 + 2 = 5 ways (As work can be completed by going by bus (A) OR by going by train (B))

Recommended Video Based on Fundamental Principles of Counting

Solved Examples Based on the Fundamental Principles of Counting

Example 1: How many ways are there to write a 4-digit positive integer using the digits 2, 3, 4, 6, and 8 if no digit is used more than once?

Solution: Since we can choose from the five available digits, we have five options for the first digit.

Similarly, because we have used up one of the digits, there are four options for the second digit and three options for the third digit.

Again 2 options are available for the fourth digit.

So the total number of ways to write a 4-digit positive integer using the digits 2, 3, 4, 6, and 8 is

$5 \times 4 \times 3 \times 2=120$

Hence, the required answer is 120.

Example 2: In a race, there are 10 participants. In how many ways can the gold, silver, and bronze medals be awarded?

Solution: The gold medal can be awarded to any one of the 10 participants, so there are 10 choices for the gold medalist.

After the gold medalist is determined, there are 9 remaining participants who could receive the silver medal. Therefore, there are 9 choices for the silver medalist.

Finally, after the gold and silver medals are assigned, there are 8 remaining participants who could receive the bronze medal. Thus, there are 8 choices for the bronze medalist.

To find the total number of ways, we multiply the number of choices for each medal: 10 choices × 9 choices × 8 choices = 720 ways.

Hence, the required answer is 720.

Example 3: The number of three-digit even numbers, formed by the digits 0,1,3,4,6,7 if the repetition of digits is not allowed, is _________

Solution:

Case 1: when 0 is at the unit's place $5 \times 4 \times 1=20$
Case 2: when 0 is not at the unit's place $4 \times 4 \times 2=32$

Total $=20+32=52$

Hence, the required answer is 52

Example 4: A class teacher wishes to assign one question from each of the two exercises in a book. If the two exercises have 20 and 15 questions, respectively, how many ways can the two questions be chosen?

Solution: Given that,

A class teacher wants to assign 2 questions from two exercises.

The first exercise has 20 questions. So, there are 20 possible ways to choose a question.

The second exercise has 15 questions. So, there are 15 possible ways to choose a question.

Thus, using the fundamental counting principle the 2 questions can be answered in $20 \times 15=300$ ways.

Hence, the required answer is 300.

Example 5: A women's fashion store sells 6 different kurtas, 5 different skirts, 2 different tops, and 4 different pairs of pants. How many different suits consisting of kurtas, skirts, tops, and pants are possible?

Solution:

Given that,

A women's fashion store sells 6 different kurtas, 5 different skirts, 2 different tops, and 4 different pairs of pants.

Thus,

$\begin{aligned} & \mathrm{n}=6 \times 5 \times 2 \times 4 \\ & \mathrm{n}=240\end{aligned}$

Hence, the required answer is 240.