Introduction to Combinations | Mathematics (Maths) Class 11 - Commerce PDF Download

Combinations
A combination is simply a manner of selecting some objects from a given set of objects in such a way that the order of their selection doesn’t matter. It is also assumed that one is not selecting a single item more than once i.e. repetitions are not allowed. Formally we may put it down as:
(n, r) or  ⁿC_r
The formula for this notation is:

where n! is the factorial of the number n, given as n! = 1.2.3….. … (n-2).(n-1).n
However, this is only valid when n>r, for physical reasons. Suppose that n<r. The term above must then represent the number of ways of selecting two objects from a set of one (i.e. n=1 and r=2, let’s say). This is not physically possible! Therefore, all the combination terms with n<r are given as ⁿC_r = 0.

The order or the arrangement of objects in a combination does not matter. It is just the selection or the inclusion of objects which is important, and not its arrangement with respect to other selected objects. Let’s clarify these concepts with a solved example.

Solved Examples

Question: A magic show has ten people in the audience. For the next act, the magician needs two people from the audience. In how many ways can he invite the two people from his audience?
Solution: What we mean by the number of ways is actually how many different pairs of people can he invite up to the stage. For e.g. suppose that we have five friends Tim, John, Robin, Alice and Sarah in the audience along with five other people.
Now, the magic trick can be conducted equally well by inviting say, John and Alice to the stage; as well as by inviting Tim and Robin to the stage. Thus, we need to find out the number of all such pairs which can lead to a success of the magic trick.
We must choose 2 people out of the total 10 people. Thus, according to the formula; we have n = 10 and r=2. Then,

It can be solved by expanding the factorial in the numerator:

= 45

Hence there are 45 ways in which the magician can select two people from his audience of ten. This is what you mean by the number of combinations of two people from a total of ten people.