Permutations and Combinations - 2023

Permutations and combinations are two important methods of arranging and grouping items. A permutation is an arrangement of items in a specific order, while a combination is a grouping of items without regard to order.

This means that the order in which the items are placed in a permutation is important, as the same items placed in a different order would result in a different permutation, while in a combination the order is not important and the same items placed in a different order would still result in the same combination.

Permutations and combinations can be used to solve a variety of problems, from counting the number of possible outcomes in a given situation to determining the probability of certain events occurring.

Permutation in Maths

A permutation in Maths is when you take a given set of numbers or objects and rearrange them in a different order. This could be done in any number of ways, depending on the number of items in the set.

For example, if you had three items in a set, you could rearrange them in six different ways. By using permutations, you can determine the number of possible outcomes in a given situation and calculate the probability of certain events occurring.

Permutation Formula:

nPr = permutation

n = total number of objects

r = number of objects selected

Let's say you have three fruits - an apple, a banana, and a cherry. You can arrange them in different orders, just like how you can arrange letters in the alphabet. For example, you can arrange them as ABC (apple, banana, cherry), or you can arrange them as ACB (apple, cherry, banana). There are six different arrangements you can make in total: ABC, ACB, BAC, BCA, CAB, and CBA.

Combination in Maths

Combination in Maths is when you combine two or more numbers or objects to create a different result than what you would get if you used the same items separately.

Combinations allow you to explore possible outcomes in a given situation and can be used to calculate the probability of certain events occurring.

Combination Formula:

nCr = number of combinations

n = total number of objects in the set

r = number of choosing objects from the set

Suppose you have four different books, A, B, C, and D. You want to choose two books to read. The possible combinations are AB, AC, AD, BC, BD, and CD.

Difference Between Permutation and Combination with Examples

Permutation Combination
Order mattersOrder does not matter
The arrangement is differentThe arrangement is the same
Example: Picking a president, vice-president, and secretary of a clubExample: Choosing 3 friends to play with
The number of arrangements is greaterThe number of arrangements is smaller
Permutation formula: n!/(n-r)!Combination formula: n!/r!(n-r)!
Repetition is not allowedRepetition is allowed
Example: The order in which colors are chosenExample: The colors chosen
nPr is always greater than or equal to nCrnCr is always less than or equal to nPr

Real Life Examples of Permutations and Combinations

Permutations and combinations can be applied to a variety of real-world scenarios like,

  1. Shuffling a deck of cards is a combination, as the order of the cards does not matter.
  2. Cooking a meal can involve permutations, as the order in which ingredients are added to a dish can affect the taste.
  3. Answering a multiple-choice test involves combinations, as there are multiple choices for each answer and the order in which answers are given does not matter.
  4. Deciding which outfit to wear for a special occasion can involve permutations, as the order in which clothing items are chosen can affect the overall look.
  5. Making a playlist for a party involves combinations, as there are many songs to choose from and the order in which they are played does not matter.
  6. Scheduling a meeting with multiple attendees involves permutations, as the order of the attendees affects who is seen first and who is seen last.
  7. Deciding which movie to watch with a group of friends involves combinations, as there are many movies to choose from and the order in which they are watched does not matter.

In conclusion, permutations and combinations are powerful methods of arranging and grouping items that can be used to solve a variety of problems. Whether it's counting the number of possible outcomes in a given situation, determining the probability of certain events occurring, or even making decisions like what to wear or what movie to watch, understanding permutations and combinations can be a valuable tool.

Test your knowledge with Upfunda Quiz!

  1. 1. Laura, Iggy, Val and Kate want to be in one photo together. Kate and Laura are best friends and they want to stand next to each other. Iggy wants to stand next to Laura because he likes her. In how many possible ways can they arrange for the photo?
  2. Ana has one 5 rupees coin, one 10 rupees coin, one 20 rupees note and one 50 rupees note. How many different values can she make with these coins and notes ?
  3. There are 13 coins in John's pocket. Each coin is either 5 or 10 cents. Which of the following cannot be the total value of John's coins?
  4. A conductor wanted to make a trio consisting of a fiddler, a pianist, and a drummer. He had to choose one of two fiddlers, one of two pianists, and one of two drummers. He decided to try each of the possible trios. How many attempts did he have to make?
  5. The "digit sum" of a whole number is the total of its individual digits; thus the digit sum of "123" is 6. How many 3-digit numbers have a digit sum of 4?
  6. Anna, Betty, Lisa and Katharina wanted to take a photo together. Anna and Katherina are best friends and wanted to stand next to each other. Lisa also wanted to stand next to Anna. In how many different ways can the photo be taken, if their wises are to be met?
  7. Ahmad walks up 8 steps going up either 1 or 2 steps at a time. There is a hole on the 6th step, so he cannot use this step. In how many different ways can Ahmad reach the top step? 

(A) 6 

(B) 7 

(C) 8 

(D) 9 

(E) 10