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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  nCr
The formula for this notation is:
Introduction to Combinations | Mathematics (Maths) Class 11 - Commerce
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 nCr = 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.

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

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,
Introduction to Combinations | Mathematics (Maths) Class 11 - Commerce
It can be solved by expanding the factorial in the numerator:
Introduction to Combinations | Mathematics (Maths) Class 11 - Commerce
= 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.

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FAQs on Introduction to Combinations - Mathematics (Maths) Class 11 - Commerce

1. What are combinations and how are they different from permutations?
Ans. Combinations are arrangements of objects where the order does not matter, while permutations are arrangements where the order does matter. In combinations, the selection of objects is important, but their arrangement is not. For example, selecting three students from a group of five is a combination, whereas arranging those three students in a specific order would be a permutation.
2. How do I calculate the number of combinations?
Ans. The number of combinations can be calculated using the formula C(n, r) = n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects being selected. The exclamation mark represents factorial, which means multiplying a number by all the positive integers less than it.
3. Can combinations be used in real-life scenarios?
Ans. Yes, combinations are widely used in real-life scenarios. For example, when selecting a group of people for a project or forming a committee, combinations are used to determine the number of different possible combinations. Combinations are also used in probability calculations and in various fields such as statistics, genetics, and computer science.
4. What is the difference between a combination and a permutation with repetition?
Ans. In a combination with repetition, the same object can be selected multiple times, whereas in a permutation with repetition, the order of the objects matters and the same object can be repeated. For example, when choosing three toppings for a pizza from a list of five, a combination with repetition allows the same topping to be chosen multiple times, while a permutation with repetition considers the order of the chosen toppings.
5. How can I apply combinations in solving probability problems?
Ans. Combinations are used in probability problems to calculate the number of favorable outcomes out of all possible outcomes. By determining the total number of outcomes and the number of favorable outcomes, you can calculate the probability of an event occurring. Combinations help in counting the number of favorable outcomes when the order does not matter.
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