Permutation and Combination are the numerous ways in which objects are selected from a set to form subsets. The problems related to permutation and combination can be considered as one of the most challenging quantitative questions you face in the GMAT exam. Hence, to solve problems from this topic, you need to be familiar with some basic GMAT Quant arithmetic permutation and combination formula.

In GMAT, the problems related to this topic can either be straightforward or complex, where you may need to solve questions based on number sequences, rolling a dice, reordering objects, etc. Though these are high-school level concepts, you should take a lot of mock tests and solve practice questions consistently, in order to master this topic as well as improve your speed and time-management skills. 

In this article, we’ll cover the basic definition of permutation and combination, and some GMAT Quant arithmetic permutation and combination formulas with examples.

What is Permutation?

Permutation — mathematical technique — can be defined as the ways in which you can arrange a certain number of objects in a specific order. Rearranging the elements of an already-structured set is also considered a process of permuting. However, while dealing with permutation, you must be careful about the selection of objects as well as the arrangement of the sets. In short, the arrangement of objects should have a definite order in permutation.

What is Combination?

In mathematics, the combination is defined as the ways of selecting certain objects from a set, where the order of selection has no importance. The combination is mainly used for solving problems related to a group of data (where the order of arrangement doesn’t matter).

GMAT Quant Arithmetic Permutation and Combination Formula with Examples:

When you find a question related to this topic, you must first figure out whether to use permutation or combination to solve the problem. Once you find that, you can easily apply the correct formula and find the answer without much difficulty. However, you have to be thoroughly familiar with the following key formulae to answer the problems, failing which, you’ll end up wasting too much time on one question. 

  • Permutation –  The number of ways in which we can permute ‘r ‘ objects from the set of ‘n’ objects without replacement and where the order matters is: nPr = (n!) / (n-r)!
  • Combination – The number of ways in which you can combine r objects from a set of n objects without replacement and where order does not matter is: nCr = n! / (n−r)!r!

Sample Questions:

  1. At a meeting of 7 Joint chiefs of staff, the chief of air staff does not want to sit next to the chief of the army. How many ways can the seven chiefs of staff be seated around a circular table?

Clearly, it is a question of circular permutation. We know that the number of ways to arrange ‘n’ items in a circular arrangement is (n-1)! 

Thus, the number of ways in which 7 people can be arranged around a circular table is: (7-1)! =  6!  

Number of ways in which 7 people can be arranged keeping the air and army chiefs  together = 5! * 2!      

Therefore, the number of ways in which 7 people can be arranged such that the 2 chiefs never be together is 6! – 5! * 2! = 5!(6-2) = 480

  1. A family consisting of one mother, one father, two daughters and a son is taking a road trip in TATA Nano. The Nano has two front seats and three rear seats. One of the parents had to drive, and two daughters refused to sit next to each other. Find the number of ways in which the seating can be done.

The driver seat can be filled by 2 people — the mother or the father.

The front seat can be taken by one of the daughters. Or both the daughters can sit at the rear. So, there are 2 possibilities here. 

  1. If one of the daughters sits in the front seat beside the driver, the front seat can be filled in 2 ways, and the rear seats can be arranged with 3!

Therefore, total no. of possibilities = 2*2*3!=24

B. If both the daughters are seated in the rear, we know that none of the daughters can take the middle seat. Therefore they have to sit at the rear window seats, which can be arranged in 2! ways. Rest of the people will be arranged by 2!. Hence total possibilities for this case is 2 *2! *2! =8

Therefore, the total number of the seating arrangements possible is 24 + 8 = 32.GMAT Quant arithmetic permutations and combinations questions can be fun and easy with consistent practice. So, make sure to practice sample questions with solved examples to be thorough with the concepts and techniques used to solve them.

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