**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.