Fundamental principle of counting

Introduction

You go shopping on a bright sunny day. You chose nice jeans and decided to pay with a credit card. But suddenly you realized that you couldn't remember your PIN. What a tragedy! Now what you can think of is to list all possible combinations to understand your pin. How many combinations are possible? If you list and count all possible combinations, the answer to this question will be difficult. In these situations, the basic principle of counting or the principle of multiplication is useful.

Number system

A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures.

Number evolution has evolved differently in different versions, including Egyptian, Babylonian, Hindu and modern American numerical. The history of enumeration development is based on mathematical evolution that is believed to have existed before the beginning of the mathematical system (Zavlatsky124).

The history of counting mathematics includes ideas for formulating measurement methods used by Babylonians and Egyptians, the introduction of pattern recognition in prehistoric number counting, and the various shapes and sizes of prehistoric people. It started with the organizational concept of numbers. Observation of natural phenomena and behaviour of the universe.

Counting

In mathematics, the key to counting quantities and finding results is the generation of a one-to-one correspondence (or bijection) of quantities with a subset of positive integers {1, 2, ..., n}. Is to be done. The basic fact that can be proved by mathematical induction is that bijection exists between {1, 2, ..., n} and {1, 2, ..., m} unless n = m. It means that it cannot be done. This fact (along with the fact that two bijections can be configured to give another bijection) is different even if the same amount is counted differently (unless an error occurs). I guarantee that it will never happen. This is a basic mathematical sentence that gives the purpose of counting. Counting (finite) quantities, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical domain of (finite) combinatorics-hence, (finite) combinations are sometimes called "count mathematics".

Origin of Counting

Numbering and counting began around 4,000 BC. In summer, it is one of the earliest civilizations. With so many people, livestock, crops and crafts in the same place, cities needed a way to organize and track everything that was consumed, added or traded.

The count started with a series of tokens.

Aryabhata's revolutionary gift to the world

Aryabhata gave the world the number "0" (zero) that made him immortal. His book, Aryabhata, astronomical and mathematical theory assuming that the Earth rotated around its own axis and gave the planet's period to the Sun (in other words, it was heliocentric).

Algebra

The algebra helps solve mathematical equations and allows the derivation of unknown quantities such as bank interest, ratios, and percentages. Algebraic variables can be used to represent unknown quantities that are combined so that the equations are rewritten.

The algebraic expression is used in our daily lives to find the distance and volume of a container and, if necessary, the selling price. Algebra is constructive in expressing mathematical formulas and relationships using letters or other symbols to represent entities. Unknown quantities of equations can be solved algebraically.

Major topics in algebra include algebraic basics, exponents, algebraic simplifications, polynomials, and quadratic equations.

Comparing numbers

Comparison of numbers is a basic idea learned in elementary school classes. If you have two numbers or magnitudes to compare, use three basic symbols.

• Equal (=)

• Greater-than sign (>)

• Greater-than sign (<)

Greater-than and smaller-than sign is used based on two numbers given for comparison.

Numbers larger than smaller numbers are expressed as − Larger number> Smaller number Numbers smaller than larger numbers are expressed as − small numbers < larger number

There are certain rules that make it easier to track numbers. compare. These rules are as follows −

• Number with more digits

• Number starting with a larger digit

Solved Examples

The boy has four shirts and three trousers. Find the total number of costumes the boy may have.

Solution −

The above question is one of the basic examples of how to count in real life. According to the question, the boy has four T-shirts and three trousers.

Therefore, the total number of costumes with boys is −

Total Costumes = 4 x 3 = 12 Boys have 12 costumes.

Consider an example where a fair die is rolled and a card is drawn from the deck. What is the total number of results in this case?

Solution −

You can find the total number of results by considering the above example as one of the examples of the actual basic counting principle.

The total number of results can be calculated as the product of the number of dice rolls and the number of cards drawn from the deck.

If the number of dice rolls is "p" and the number of cards drawn from the deck is "q", the total number of results is calculated as p×q.

A beautiful cube has six sides. Therefore, the total number of results for dice is p = 6

There are 52 cards in the deck. Therefore, the total number of possible consequences when a card is drawn is q = 52.

The total number of results when both events occur at the same time is as follows −

$\mathrm{p\times\:q\:=\:6\times\:52\:=\:312}$.

Conclusion

For simple counting, the product or product rule is the basic principle of counting. Simply put, the idea is that there is a way to do something, a way to do something else, and a way to do both. Counting can be defined as the act of determining the number or total number of objects in a set or group. Simply put, this means saying the numbers in order and assigning values to the elements in the group based on a one-to-one correspondence.

FAQs

1. What are the basic principles of counting?

Ans. If a event occurs in different ways in m and another event can occur in different ways in n, the total number of events that occur is m × n.

2. Suppose you have 3 pairs of shoes and 4 pairs of socks. How can you wear them?

There are 3 pairs of shoes and 4 pairs of socks. You can carry, or 3x4 times, or 12 times.

3. What does the Basic Principle of Counting (FPC) mean?

The Basic Counting Principle (FPC) is a way to find the number of possible outcomes in a particular situation.

4. What is the term for count?

The terms of count are

• Addition Principle

• Multiply Principle

5. Assuming you go out and buy candies and your bakery has 15 cupcakes, 20 donuts and 13 muffins. If you are asked to choose delicious sweets, how many candies can you choose?

Here we used the basic count addition principle. You have to choose between cupcakes, donuts, or muffins. Therefore, you can choose from 15 + 20 + 13 = 48 treats.