Meaning of P (n,r) and C (n,r)

The topic of permutation and combination is an important part of mathematics. Permutation and combination are two different ways to represent a class of objects. Both permutation and combination are different concepts of math and most of the students get confused between them. In combination, the order arrangement doesn’t matter but in the case of permutation the arrangement of orders matters. Simply put, an ordered combination is called a permutation. This topic is one of the most important ones for IIT JEE as well as other competitive exams. It carries almost 20-25 marks in IIT JEE Examination. So, do not skip this chapter as it can help you secure a decent rank in your exam.  

What Is Permutation?

A permutation is a mathematical process that ascertains the number of possible arrangements in a set. In this case, the order of arrangement matters. If the set is already in a particular order, the rearrangement of the elements is called a permutation.

Most of the mathematical problems include selecting only various objects from a set with a particular order. This takes place in important parts of mathematics, especially probability. Permutations help us in prominent ways to solve problems that arise when various orderings on certain finite sets are taken into consideration.  

Formula of Permutation

The Permutation Formula that we use is expressed in the following way:

P(n,r) = (n!) / (n-r)!

Here, n represents the total number of objects that are present in a set. 

And r represents the number of selected objects arranged in a certain order. 

The factorial sign ‘!’ represents the product of all positive integers less than or equal to the number preceding the sign. 

We use the permutation formula in situations when we want to choose various objects from a set of objects and arrange the selected objects in a certain order. 

Types of Permutation

There are two types of permutations that occur in mathematical problems. They are: 

Repetition: When some of the chosen objects are identical, the problem is transformed into permutations with repetition. This process is used when we are asked to make different choices each time and with different objects. 

No Repetition: Suppose you are in a race, and the first three people are considered. Now, no one can be first and second at the same time. When we are required to reduce 1 from the previous term each time, we apply this method. 

What is Combination?

Combination is a method of choosing things from an assortment where the request for choice doesn’t make any difference. Assume we have a bunch of three numbers A, B, and C. Then, at that point, the number of ways we can choose two numbers from each set is characterized by combinations. 

Combinations are determinations made by taking a few or all of various items, independent of their courses of action. In modest cases, it is feasible to count the number of combinations, yet for the cases which have an enormous number of gathering of components or sets, the chance of having a large number of combinations is very high. 

Formula of Combination

The combination formula that we use to solve problems is stated below:

C(n,r) = n!r! ×n-r! 

Here the, n represents the total number of elements in a set and r represents the selected objects, same as the permutation formula. 

Types of Combination

There are two types of Combinations that are taken into consideration. They are: 

Repetition: If the two combinations have the same elements repeated the same number of times, regardless of their order, they are considered identical. 

No Repetition: In this case, we can take lottery numbers as an example. If we have to select any lottery number from a bunch of numbers, there won’t be any repetition.

Difference Between Permutation and Combination

There are a lot of differences between Permutation and Combination. They are as follows:

• We use permutations for lists where the order matters. Whereas combinations are used for groups. In the case of combination, the order does not matter.
•   In permutation, we select r objects from a set of n objects in a certain order. But for combination, it is about the number of possible combinations of r objects from a set of n objects.
• Permutation is about the arrangement of elements whereas combination is the selection of elements from a set.
•   Permutations are determinations made by taking a few or all of various items, independent of their courses of action. Whereas, a solitary combination can be gotten from a solitary permutation.
•   A permutation is defined as ordered subjects. The combination is defined as unordered sets. 

Uses of Permutation and Combination in Daily Life

Apart from the mathematical uses, Permutation and Combination also have various uses in our day-to-today life. Some of the most common uses are as follows:-

Lottery Number: It is an example of a combination, in the lottery game, different numbers are elected. If a person has to select 5 numbers from the list of the first 15 natural numbers. He must have to select digits in a non-repeating manner.

Vehicle Plate Number: This is an example of Permutation. In the number plate of vehicles, the first 2 digits show the code of state like RJ for Rajasthan, UK for Uttarakhand, etc. the next shows the vehicle type like T for Taxi, C for car, and the last 4 digits are permuted & are different for every vehicle. 

Examples

Permutation example:

Ram has 4 chairs and he wants to place 3 dolls on these chairs. In how many possible ways can he do this? 

Solution: Given n = 4 and r = 3 

Applying the permutation formula of          

P(n,r) = n!n-r!                                                     

We get, the possible ways = 4!4-3! =  4 ×3×2×11= 24 

So, the answer is 24 possible ways he can place 3 dolls on 4 chairs. 

Combination Example:

How many different combinations do you get if you have 4 items and choose 2?

Solution: Insert the given numbers into the combinations equation and solve. “n” is the number of items that are in the set; “r” is the number of items you’re choosing (2 in this example):

Now from the Combination Formula, we can get

C(n,r) = n!r! ×n-r! = 4!2! ×4-2! = 4!2! ×2! = 4×3×2×12×1×2×1 = 244 = 6

The solution is 6.

Conclusion

As we have noticed above, Permutation and Combination is one of the important topics for examination and it is also associated with our life. In the last year, 5 questions were asked from this chapter in IIT JEE Mains as well as in Advance paper.  Many Students have confusion in Probability and permutation & Combination. 

If we consider the ratio of the number of required subsets to the number of all probable subsets, the theory of probability was introduced. For a one-word explanation regarding permutation and combination, permutation is arrangements and combination is selections. Do not skip this important topic and go through the notes thoroughly.