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Permutations and Combinations

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

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.
  • Find the rank of the word “BANDAGE”,if words are arranged in the order as they appear in a dicitionary.
  • Find the number of ways of distributing 10 different pens to 4 students if the order in which the pen is given also matters.Also find the number of ways if the pens are all identical.
  • There are 2 professors Arvind and Jatin incharge of choosing boys and girls for a particular competition from their respective branch of study.If professor Arvind chooses 3 girls and 5 boys from his branch and professor Jatin chooses 2 girls and 6 boys from his branch.In how many ways can a group of only 8 people be made for the competition consisting of atleast 2 girls.
  • Let us consider there are 6 different balls named as ball 1,ball 2,....ball 6 alloted positions 1,2,3,4,5,6.Find the no. of ways in which the arrangement of these balls can be done if exactly three among these 6 balls occupy their numbered positions and the rest 3 don't.
  • If there are n sides in polygon 1 and m sides in polygon 2 and if m+n=19.If then sum of number of diagonals of polygon1 and polygon 2 is 62.Find the values of m and n.

...And many more challenging and interesting questions!

This tutorial 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.