2-COMBINATIONS

We are often interested in determining the number of different groups of r objects

that could be formed from a total of n objects. For instance, how many

different groups of 3 could be selected from the 5 items A, B, C, D, and E?

To answer this question, reason as follows: Since there are 5 ways to select

the initial item, 4 ways to then select the next item, and 3 ways to select

the final item, there are thus 5 · 4 · 3 ways of selecting the group of 3 when the order

in which the items are selected is relevant. However, since every group of 3—say, the group 

consisting of items A, B, and C—will be counted 6 times (that is, all of the permutations

ABC, ACB, BAC,BCA, CAB, and CBA will be counted when the order of selection is relevant),

it follows that the total number of groups that can be formed is 

In general, as n(n − 1) · · · (n − r + 1) represents the number of different

ways that a group of r items could be selected from n items when the order

of selection is  relevant,  and  as  each group of r items will be counted r!

times in this count, it follows that the number of different

groups of r items that could be formed from a set of n items is  

Notation and terminology

We define  for 𝑟≤𝑛, by

and say that represents the number of possible combinations of n

objects taken r at a time. It can be noted as 

Thus, represents the number of different groups of size r that could

be selected from a set of n objects when the order of selection is not considered relevant.

A committee of 3 is to be formed from a group of 20 people. How many

different committees are possible?

There are  = 1140 possible committees.

From a group of 5 women and 7 men, how many different committees

consisting of 2 women and 3 men can be formed?

As there are  possible groups of 2 women, and possible groups

of 3 men, it follows from the basic principle that there are =350

possible committees consisting of 2 women and 3 men.