The entire concept of math started with the necessity of counting. Our ancestors used to count using buttons, beads, sticks, and even stones.
However, they soon realized these counting methods were inefficient for counting large quantities and sums. This gave rise to one of the most critical concepts in math – the fundamental counting principle.
The fundamental counting principle is used for determining large counts and probabilities quickly and efficiently. In addition, it is used in probability problems to determine how different events can occur.
The fundamental counting principle, also known as the fundamental counting rule, is used to figure out the different possible outcomes for a situation.
There are different counting principles: addition, subtraction, division, cardinality and multiplication. However, the one principle that is used in the case of fundamental counting is multiplication.
This basically states that if there are p ways to do a certain task and q ways to do another, there are many pxq ways to both tasks.
Now that you know the basics of the fundamental counting principle, let's look at some examples of it.
You have 3 shirts, namely A, B, and C. You also have 4 pairs of trousers, namely x, y, z. Then the number of possible combinations of outfits that you will have is:
3x4 = 12
Suppose you roll dice with 6 sides and draw a card from a deck of 52 cards. Then the total possible combination of your experiment can be:
6x52 = 312
There are 8 newspapers and 5 weekly magazines in Chicago. If a person wants to subscribe to one daily newspaper and one magazine, then the choices present are:
8x5 = 40
A restaurant has 4 types of sandwiches, 5 sides, 3 desserts and 6 drinks. So, the total number of meal combos that is possible are: 4x5x3x6 = 360
There is a survey that you take with 6 yes and no answers. So, the total number of ways in which you can solve it are:
2x2x2x2x2x2 = 64
Let us now take a look at some intermediate examples.
John wants to go to Denver. First, he can choose from 3 types of bus services or 2 types of trains to reach Chicago. From there, he can choose from 2 types of bus services or 3 trains to head to Denver. Thus, the total number of ways that John has to head to Denver are:
John can take 3+2 types of services to head to Chicago. From there, he has another 2+3 ways to head to Denver.
So, the total no. of ways to travel are:
5x5 = 25
The formula for calculating the factorial (!) of any number is:
N! = n.(n-1).(n-2)…..
So, if we have to find out the factorial of 5! then the process will be:
5! = 5.4.3.2.1 = 120
Permutation refers to the different ways in which a group of things can be ordered.
The formula for permutation is shown below.
The combination is used for selecting a distinct group of different objects without particular order.
The combination formula is given as follows:
The fundamental counting principle is an arithmetic concept that allows us to determine the choices and combinations among groups. It helps us understand the different ways in which events can occur.
The principle states that the number of outcomes of an event is the product of outcomes of each different event.
The product of the events helps us understand the total outcomes that can occur.
The basic formula for the fundamental counting principle is:
Events = p, q, r
Thus, the total number of outcomes = pxqxr
Moving on, let’s look at the use of this principle for determining the sample space.
The term sample space means all the possible outcomes for a particular event. For example, your outcomes will be either heads or tails when you toss a coin. Thus, your sample space for your toss of a coin is heads and tails.
When you toss a dice, you can either get 1 or 2 or 3 or 4 or 5 or 6.
Thus, your sample space for rolling dice is 1, 2, 3, 4, 5, 6.
To determine the sample space for an event, we can use the fundamental counting principle.
For example, if we have to flip a coin and roll a dice, the total possible outcomes will be:
H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6.
However, when we use the fundamental counting principle, which states that the total outcomes are equal to the products of individual events, we can easily find out the total number of outcomes without listing them.
Thus, the total outcome of rolling dice is 6
The total outcomes for flipping a coin is 2
So, the total sample space can be calculated by multiplying the outcomes of both events.
So, the sample space here is 2x6 = 12.
Ans. The fundamental counting principle works for multiple events by multiplying the outcomes of each event. However, in the case of permutation, you can use this method for counting the different ways to choose and arrange several given objects. Thus, both methods have different applications.
Ans. The combination formula used the letter r. r here stands for the number of different chosen items for one instance. This helps us find the different combinations of events that can occur. Thus, when applying the combination formula, make sure that you enter the value of r correctly.
Ans. The fundamental counting principle helps us understand how selections are done based on the number of available choices. It helps us determine the selection process based on the number of choices that we have.
Moreover, the principle simplifies the selection process and helps us calculate other probability choices. The principle is also based on the knowledge of numbers and finds application in a vast range of real-life situations.
Ans. The three principles of counting are:
This involves using a list of words to count repeatedly. The ordered list of countable items must equal the counting words used.
This involves understanding that each object or item in a group can be counted only once. This is useful in the early stages of a child’s educational development when they first learn to start counting.
This method involves understanding that the last number in a group used for counting objects represents the number of items in that group.
Ans. nCr represents the different selection of objects from a particular group of objects. The order of the objects present does not matter here. nCr is also called combination and is a great method for object selection. The formula for nCr = nC(n-r). Here, r number of objects is chosen from n number of objects, and the order of objects is not important.
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