What is Best way to Practice Permutations and Combinations?

What is Best way to Practice Permutations and Combinations?

The best way to practice permutations and combinations is to practice as many questions as possible.

The first step is to know the basics of how both of them functions. For example, permutation is used to arrange objects, and combinations is used to select objects.

When you begin to solve question, give enough time to the questions as this will aid in understanding the question and then you can discover your own way of solving it.

If the question has huge numbers like 100 balls or 50 people, then reduce these numbers to smaller numbers and find a way to solve the problem.

Permutations and combinations

Permutations and combinations can be defined as the principles of counting and it is applied in several situations. A permutation can be defined as a count of the different arrangements which can be created from the given set of things. The details, order or sequence are important in permutation. The names of three countries can be written as [USA, Brazil, Australia] or [Australia, USA, Brazil] or [Brazil, Australia, USA]. The sequence or the order in whichthe names of the countries are written is crucial. In combinations, the names of three countries are simply a group and the sequence or order is not important.

What are permutation and combination?

Permutation and combination are the ways used in counting how many outcomes are possible in various situations. Permutations are known as arrangements and combinations are known as selections. According to the  fundamental principle of counting, there are sum rules and the product rules to be used in counting easily.

Let us take us an example where there are 14 boys and 9 girls. If a boy or a girl is to be chosen as the monitor of the class, the teacher can choose 1 out of 14 boys or 1 out of 9 girls. The girl can do it in 14 + 9 = 23 ways by making use of the sum rule of counting. Let us look at another way. There is a boy named Sam who has one main course and a drink.  Now he has different options like pizza, hot dog, burger, watermelon juice and orange juice. What can be all the different possible options that he can try?  There are3 drink choices and 2 drink choices. These two choices are now multiplied, 3*2 = 6. Hence, Sam has 6 combinations which he can try by using the product rule of counting. This can be explained by the given diagram below.


To understand permutation and combination in a better way, we must bring in the concept of factorials. The product of the first n natural numbers derived is n. The number of ways of arranging n unlike objects which we get is n.


A permutation can be defined as an arrangement in a definite order of a number of objects taken few or all at a time. Let us take an example where there are 10 numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  The number of different 4-digit- PIN which can be created by using these 10 numbers is 5040, P(10,4) = 5040. This is a simple example of permutations.


A combination talks about grouping. The number of groups that can be formed from the available things can be calculated by using combinations. Now let us understand this with a simple example.  A team of 2 can be formed from 5 students (Wiliam, James, Noah, Logan, and Oliver). The combinations can be made in the following 10 ways by which the team of 2 can be made.

  • William James
  • William Noah
  • William Logan
  • William Oliver
  • James Noah
  • James Oliver
  • James Noah
  • Logan Noah
  • Logan Oliver
  • Oliver Noah

The above is a very simple example of combinations.

C (5,2) = 10.

Can you solve this question?

Q:How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed?

Permutation and combination formulas

Permutation and combination formulas are useful to take out the permutation and combination of r objects taken from n objects.  The concept of permutations is used for finding the different arrangements and the concept of combinations are used for finding the different groups. When the values of n and r is given, the permutations are always greater than the combinations.

Permutation and combination formulas

The counting situations are studied to find out whether to use permutations or combinations. In this way, the permutation and combination formulas are used.

Formula 1

Factorial of a natural number n

N! = 1 * 2* 3 * 4 *………….*n

Formula 2

The number of distinct permutations of r objects which can be derived from n distinct objects is

Formula 3

The number of permutations of n different things, taken r at a time, where repetition is allowed, is

Formula 4

Formula 5

Formula 6