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Permutations and Combinations

A Tutorial on Permutations and Combinations: Notes, Figures and Problems with Solutions

They might be of use to Indian students preparing for the ISC or CBSE Class 11 and 12 Examinations, IIT JEE (main and advanced), AIEEE; students from across the world preparing for their A Level Examinations, IB (International Baccalaureate) or AP Mathematics.


This compilation of notes has been prepared by Ayushi Patel of IIT Gandhinagar.

Complete Tutorial with Problems, Figures and Solutions : 





Summary and Outline of Notes: Here's a quick recap of the outline of topics which were covered in this tutorial


  • Permutation and Combination

  • Fundamental Principle of Counting:

  • Rule of product

  • Rule of sum

  • Circular permutations

  • Division and distribution:The number of ways in which n different objects can be divided in m groups

  • Multinomial theorem


Here are a few of the problems we learn how to solve in the tutorial above:

  • Find the number of ways in which 10 different chocolates can be given to 4 persons?Also find the number of ways if exactly 2 chocolates should be given to person 1.
  • Find the number of factors of the number 680400.
  • Find how many 4 letter words can be formed from the word “POSITIVE” ?Also find the no. of words  if there should be atleast 3 consonants in the word and they are together.
  • If there are 11 bus stops and 6 persons are needed to be left at the bus stops .In how many ways can the persons be left at the bus stops?
  • Suppose there are 9 total subjects available for students such that each student is required to take atleast 5 subjects.In how many ways can this be done?If among the 9,3 are compulsory subjects in how many ways can they choices be made?
  • If there are total of n teams in a basketball competion and there are total of 72 matches played such that every team needs to play with every other team.Find the value of n if 3 teams backout after playing 4 matches.
  • If 8 places are allotted to 8 officials such that each place is reserved by their name.Find the number of arrangements such that each official is seated such that he is not in the place reserved for him.
  • If there are 12 identical tasks to be done and there are 3 workers and the task can be distributed such that each worker can do any number of tasks. Find the number of ways in which the tasks can be allotted. Also ,find the number of ways in which the tasks can be allotted if the 12 tasks are different.

...And many more challenging and interesting questions!