GMAT Quant: Arithmetic â€“ Permutation & Combination

Permutation & Combination is a method of counting the number of ways of selection and arrangement. In GMAT, the problems might be one liner of can be a complex one where you have to make cases as well. Let us look at some of the questions from this topic:

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 staffs 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)!

Number of ways in which seven people can be arranged is 6!Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (1)

Number of ways in which seven people can be arranged keeping the above two guys together = 5!. 2!Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (2)

Therefore number of ways in which seven people can be arranged such that the above people 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. One of the parents has to drive, and two daughters refused to sit each other. Find the number of ways in which the seating can be done.

The driver seat can be filled with two possibilities. Here we will make 2 case

a.If one of the daughters is sitting in the front seat beside the driver.

Therefore the front seat can be filled with two possibilities, and the rear seats can be arranged with 3!

Therefore , total possibilities = 2*2*3!=24

b.If both the daughters are seated in the rear. Clearly, none of the daughters can take the middle seat. Therefore they have to sit at the rear window seat. Which can be arranged by 2!. 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 arrangement possible is 24 + 8 = 32.

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