Combination Formula

Combination Formula

The combination formula is used widely in mathematical problems to find out the number of ways in which a particular item can be selected from a collection of objects.  In the combination formula, the order of the selection does not matter that much, but the selection process of the objects from a larger group is important. 

Through this Formula One can find all the possible combinations by taking a subset of items. All the different possible solutions can be obtained through this formula. Combination and permutation are used closely to find out the number of different ways in which a subset can be selected from a large group.

Day-to-Day Examples Where Combination Problems are Used

We use a combination formula in different cases like calculating the number of lottery tickets because it has different combinations. Other examples include linear combinations, number of ingredients, musical chords, etc. 

The Formula for Solving Combination Problems

The following formula is used to find out the number of combinations. ‘N’ represents the total number of items and r is the desired number of elements that can be obtained. 

nCr = n!/ (n-r)! r! 

Another formula that can be used to find out the combination formula when permutation is given is : 

C(n,r) = P (n,r)/ r! 

Here r denotes the size which is usually greater than or equal to n. 

N is used to represent the total number of items in the set. 

And ! (exclamation sign in English) is used to represent factorial operator which means that we multiply all the numbers preceding the no. 

Remember, r and N will always be non-negative. 

Solved Examples

Question 1: If a basket has 18 different items and Samir is asked to choose only four items from the table basket, how many different combinations are possible in this situation?

Ans: In this case, N is equal to 18 denoting The total number of items and r is equal to four (the desired number from the set). 

Therefore C(18, 4) = 18C4= 18!/ (18- 4)! 4! 

= 18* 17* 16* 15* 14!/ 14! 4* 3 *2*1 

= 3060 

Hence, the number of ways in which items can be picked from the basket is equal to 3060. 

Question 2: In an interview, five people appeared, namely ABC D&E. Only three people have to be selected by the panel. Find out the number of ways in which the selection can be done.

Ans: here, n= 5 and r=3. Since the total number of combinations is required, we use the formula, 

nCr = n!/ (n-r)! r! 

Substituting n with 5 and r with 3, we get 

5Cr3= 5!/ (5-3)! 3! 

= 5*4*3!/ 2*3!

= 20/2= 10

Hence there are ten ways in which three out of five candidates can be selected by the panel.