CHAPTER 3 - Introduction to Combinations

In the previous section, we encountered permutations, which correspond to arrangement of objects /things/ entities. In this section, we will encounter combinations which correspond to selection of things (and not their arrangement). We do no intend to arrange things. We intend to select them. For example, suppose we have a team of  cricket players. We intend to select a playing team of  out of these  players. Thus, we want the number of ways in which we can select  players out of  players. (We are not interested in arranging those  players in a row – only the group/ combination of those players matters).

Let us make this concept more specific. Suppose we have a set of  letters. In how many ways can we select a group of  letters from this set? Suppose we had to find the number of arrangements of  letters possible from those  letters. That number would be . Consider the permutations that contain the letters  and . These are  in number, namely  and .

Now, what we want is the number of combinations and not the number of arrangements. In other words, the  permutations listed above would correspond to a single combination. Differently put, the order of things is not important; only the group/combination matters. This means that the total number of combinations of  letters from the set of  letters available to us would be  since each combination is counted  times in the list of permutations. Thus, if we denote the number of combinations of  things taken at a time by , we have

In general, suppose we have  things available to us, and we want to find the number of ways in which we can select  things out of these  things.

We first find the number of all the permutations of these  things taken  at a time. That number would be . Now, in this list of  permutations, each combination will be counted  times since  things can be permuted amongst themselves in  ways. Thus, the total number of combinations of these things, taken  at a time, denoted by , will be

You should now be able to appreciate the utility of the fundamental principle of counting. Using only a step-by-step application of this principle, we have been able to obtain an expression for  As we progress through the chapter, you’ll slowly realise that each and every concept that we discuss and each and every expression that we obtain follows logically as a consequence of this simple principle.

Example: 1

Consider the word . (a) In how many ways can we select a pair of letters consisting of a vowel and a consonant ? Both vowels? Both consonants? (b) In how many ways can a  – letter selection be made with more than  vowels?

Solution: 1-(a)

We have  vowels and  consonants available in the word . A vowel can be selected in  ways while a consonant can be selected in ways. Thus, a pair consisting of a vowel and a consonant can be selected in  ways. Two vowels can be selected in  ways while two consonants can be selected in  ways.

Solution: 1-(b)

Since we want more than  vowels, we could have  or  vowels in our -letter selection. Suppose we select  vowels for our -letter group. This can be done in ways. The two remaining letters (consonants) can be selected in  ways. Thus, -letter groups containing  vowels can be formed in ways. Similarly, if we had  vowels, the possible number of groups would be. There’s only  group possible containing (all) the  vowels. Thus, the number of required selections is Now that we’re done with the introductions, lets move on and see some really interesting and diverse applications of the basic concepts covered till now.