Permutation Combination and Probability - Sampling with / without replacement


These two topics are invariably considered by many GMAT aspirants to be tougher than a whole lot of other topics in the quant section of the GMAT test.

The biggest advantage with a Permutation Combination or a Probability question is that many a times you will not have the need to do any calculation at all. Most answers will be either mentioned in terms of factorials or as 3C2 or 4P3 format.

Having said that, the main catch with these two topics is that you need to think a little differently while attempting questions from these two topics as compared to say a question from profit and loss or speed time and distance. Once you have managed to figure that difference, it is one of the easiest topics to work on.

The common question that many of my students seem to be coming back to me with is whether the given question is a combination question or a permutation question?

Let me see how that can be resolved with the help of the following examples on permutation and combination.

To begin with let us introduce the term sampling. A sampling is an activity of picking up (selecting) none or more or all items from a group of items.

For e.g. let us consider the act of selecting 3 students from a class of 10 students to attend a seminar. This act of selecting 3 students from 10 students is commonly referred to in the Permutation Combination (Combinatorics / counting methods) parlance as sampling.

Sampling can be classified broadly in terms of two parameters.

1. Sampling with replacement or sampling without replacement
2. Sampling with ordering or sampling without ordering (arrangement or without arrangement)

In the example provided above, we have to choose 3 different students from 10 students. After the 1st student is selected, we have only 9 students in the class to choose the 2nd student. After the 2nd student is selected, we have only 8 students in the class to choose the 3rd student. In effect, we had done 3 samplings - 1 for choosing the 1st student, 2 for choosing the 2nd student and the last one for selecting the third student.

Well as you would have noticed, in the above example the number of students available after each round of sampling kept decreasing. i.e., while sampling the second time, we only had 9 students from whom we could select one student. This kind of sampling where we did not replace the outcome of the earlier sampling back is know as "Sampling without replacement".

Let us take a look at another example where we will replace the outcome of the earlier samplings back so that it (they) is (are) available for the subsequent stages of sampling. Let us say we are interested in writing a 3-digit positive integer. The first place or the hundreds position can take any of the 9 values from 1 to 9 (it cannot include "0"). The second position or the tens place can take any of the 10 values 0 to 9 and the third place or the units digit can be any of the 10 values 0 to 9.

Take for instance, a number like 242. The digit "2" appears both in the hundreds as well as the units place. Writing the 3 digit number actually involved 3 samplings. The first for the hundreds place, second for the tens place and finally for the units place. As the number 2 was available to be used again when the sampling was done for the tens and the units place, this is an example of sampling with replacement.

Similar examples can be found while forming words using the alphabets of a language. Take for example the word "Permutation". The letter "t" has been used twice. It is again an example of Sampling with replacement.