Combination Formula Explained

In our day to day life, we are facing many situations in which we have to make the selection of some objects taken from a collection. For example, when we select 3 bells from a set of 10 bells in all possible orders. We can compute these with the help of Permutation and Combination. In Mathematics as well as in statistics, Combinations are very useful for many applications. In this article, we are going to discuss the concepts of Combinations with a Math Combination formula explained.

What is the Combination: Combinations are the various ways in which objects from a given set may be selected. Normally it is done without replacement, to form the subsets. Combinations are a way to find out the total outcomes of an event where the order of the outcomes does not matter.

Thus the Combination is the different selections of a given number of objects taken some or all at a time. For example, if we have two alphabets A and B, then there is only one way to select two items, we select both of them. For Combinations, k elements are selected from a set of n objects to produce subsets without bothering about ordering. Here Combinations BA and AB will be no longer distinct selections. Thus by eliminating such cases we get only 10 different possible groups, which are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.

The Understanding What is the Combination Using a Few Examples:

There are also two types of Combinations (remember the order doesn't matter now):

When Repetition is Allowed: Let us take the example of coins in your pocket (5,5,5,10,10)When no Repetition: Let us take the example of lottery numbers, such as (2,14,15,27,30,33)

1. Combinations with Repetition.

2. Combinations without Repetition.

This is how lotteries work. We draw the numbers one at a time, and then if we have the lucky numbers (no matter in what order) we win the lottery!

The easiest way to explain it is to:

Assume that the order does matter (ie Permutations),

Then alter it so that the order or sequence does not matter.

Let’s see a pool ball example, let's say that we just want to know which of the three pool balls are chosen, and not in the order of the sequence.

We already know that 3 out of 16 gives us 3,360 Permutations.

But many of those are the same for us now because we don't care about the order!

Combination Equation/Combination Formula Explained in Maths

The Combination equation is ⁿC_k can be known as counting formula or Combination formula explained in Maths. This is because these can be used to count the number of possible Combinations in a given situation.

In general, if there are n objects available. And out of these to select k, the number of different Combinations possible is denoted by the symbol ⁿC_k.

The number of subsets, denoted by ⁿC_k, and read as “n choose k”. will give the Combinations. It is obvious that this number of subsets has to be divided by k! , as k! arrangements will be there for each choice of k objects. 

Thus, the Combination equation.

\[^{n}\textrm{C}_k=C(n,k)\]

= \[\frac{n!}{(n-k!)\times k!}\]

Notations in ⁿC_k Formula

• k is equal to  the size of each Permutation

• n is equal to the size of the set from which elements are permuted

• n, k can be defined as non-negative integers

• ! is known to be the factorial operator

The Combination formula in Maths shows the number of ways a given sample of “k” elements can be obtained from a larger set of “n” distinguishable numbers of objects.

Hence, if the order doesn’t matter then we have a Combination, and if the order does matter then we have a Permutation. Also, we can say that a Permutation is an ordered Combination.

To use the Combination formula we have discussed above, we will need to calculate the factorial of a number. A factorial of any number can be defined as the product of all the positive integers which is equal to and less than the number. A factorial symbol can be denoted by an exclamation point (!). For example, to write the factorial of 4, we will write 4!.

Let’s calculate the factorial of the number 4,

4! = 4 × 3 × 2 × 1i.e. 4! = 24

Questions to be Solved

Question 1: In a lucky draw of ten names are out in a box out of which three are to be taken out. Find the number of total ways in which we can take four names out of the given box.

Solution: Here, we will take out three names from the box. Thus, the selection is four without having to bother about ordering the selection.

Thus, the possible number of ways for finding three names out of ten from the box can be written as:C (10, 4) which is equal to

= \[\frac{10!}{(10-4!)\times 4!}\]

= \[\frac{10\times 9\times 8\times 7\times 6\times 5\times 4}{6\times 5\times 4\times 3\times 2\times 1\times 4\times 3\times 2\times 1}\]

= 210

Question 2: Mother asks her daughter to choose 4 items from the drawer. If the drawer has 18 items to choose, how many different answers could the daughter give?

Solution: Given, k = 4 (item subset)

n = 18 (larger item)

Therefore, simply: find “18 Choose 4”

We know that, Combination = C(n, k) = \[\frac{n!}{k!(n-k)!}\]

= \[\frac{18!}{4!(18-4)!}\]

= \[\frac{18!}{14!\times 4!}\]

= 3,060 possible answers.

Difference Between Permutation and Combination

Combinations mean a selection of r things out of n things and Permutation means the arrangement of r things out of n things. For the given r things from the given n things, the number of Combinations is less than the number of Permutations. 

In Combination, we do not care about the order and the things can be taken in any order but in Permutation, we care about the order and the things are arranged in a particular order. 

Important Points to Remember

Students must remember some important points related to Combination and Permutation. The most important points to remember in the topic include the following:

• For the phrase “number of Combinations”, you must remember “number of selections”. When you have to select things, you can take them in any order. For example, ABC and CBA have different arrangements but they have the same selection. 

• The Combination formula is based on the fact that in the list of all Permutations of r size, each selection is counted r! times. 

• If there are n objects, the number of ways of Combinations 0 or n objects is 1. The number of ways of selecting 1 object or (n-1) object is n.