Combination Formula

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, selecting 3 balls from a set of 10 balls 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 will see the concepts of combinations with a math combination formula. Let us start learning the concept.

Combination Formula

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, AB and BA 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.

Combination Formula

The formulas nCk is popularly known as counting formula. 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 nCk.

The number of subsets, denoted by nCk, 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,

nCk = C(n,k)= \( \frac{n!}{(n-k)!× k!}\)

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 a combination formula, we will need to calculate a factorial. A factorial is the product of all the positive integers equal to and less than the number. A factorial symbol is an exclamation point (!). For example, to write the factorial of 4, we will write 4!. To calculate the factorial of 4,

4! = 4 × 3 × 2 × 1

i.e. 4! = 24

Solved Examples on Combination Formula

Q: 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.

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

Thus, the possible number of ways for finding three names out of ten from the box will be :

C (10, 4)

= \( \frac{10!}{(10-4)! ×4!} \)

= \( \frac{10 × 9 × 8 × 7}{(4 × 3 × 2 × 1)} \)

=  210