Permutations and Combinations

Combinations

Combinations: Have you ever been offered a bag full of different types of chocolates and asked to take three chocolates out of them? I’m sure you take a moment to think about the total number of ways in which you can take out the three possible chocolates and then choose the one that corresponds to you picking out your favourite ones.

That’s what we are going to deal with mathematically here. The total possible ways in which you could have selected the three chocolates are actually the total number of combinations possible for the three chocolates taken from the bag! Let us know more about this concept!

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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:

nC= \( \frac{n!}{r!(n-r)!} \)

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.

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Combinations

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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.

Combinations

(Image Source: Wikimedia)

Solved Examples for You

Question 1: 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?

Answer : 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,

10C= \( \frac{10!}{2!(10-2)!} \)

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

= \( \frac{10.9.8!}{2!.8!} \) → \( \frac{10.9}{1.2} \)

= 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.

Question 2: What are the combinations?

Answer: A combination refers to a manner of choosing some objects from a certain set of objects in such a way that the order of their selection does not matter. There is also an assumption that one is not selecting a single object more than once because it does not allow repetitions.

Question 3: What is the difference between combination and permutation?

Answer: The difference between combinations and permutations is the ordering in them. In permutations, only the order of the elements matter. While, in combinations, it does not matter. For instance, if your locker “combo” is 3784 and you enter 4873 into your locker, you won’t be able to open it because it is a different ordering which is its permutation.

Question 4: How many types of combinations are there?

Answer: There are three types of combination reactions. First one is a combination of two elements. Next, we have a combination of two compounds and finally the combination of an element and a compound.

Question 5: Does order in combination matter?

Answer: The order of objects in a combination does not matter. It is merely the selection or the insertion of objects that is important, and not the arrangement of them with regards to the other selected objects.

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