Combination Formula
"Combination" in math means "selection". The combination formula is used to find the number of ways selecting k things out of n things. Sometimes we prefer the selection where the order doesn't matter. For example, if we have to choose a team of 3 members out of 15 members, we just have to select 3 members out of 15, but we don't need to order them. In such cases, we apply the combination formula to find the number of ways of doing it. Let us learn the combination formula along with a few solved examples.
What is Combination Formula?
The combination formula is also known as "n choose k formula" which is denoted by C (n, k) (or) \(n_{ C_{k}}\) (or) \( _{n} C_{k}\) (or) \(\left(\begin{array}{l}n \\k\end{array}\right)\). It is also known as a binomial coefficient. It is used to find the different number of ways of selecting k different things out of n different things. This formula involves factorials.
The n choose k formula is:
C (n , k) (or) \(n_{ C_{k}}\) (or) \( _{n} C_{k}\) (or) \(\left(\begin{array}{l}n \\k\end{array}\right)\) = \(\dfrac{n!}{(nk)! k!}\)
Solved Examples Using Combination Formula

Example 1
Find the number of ways of forming a team of 3 members out of 15 members.
Solution:
Here, the total number of members is n = 15.
The number of members to be selected to form a team is k = 3.
This can be done in C(n , k) ways and we will find this using the combinations formula.
C (n , k) = n! / [ (nk)! k! ]
C (15, 3) = 15! / [ (15  3)! 3! ]
C(15, 3) = (15 × 14 × 13 × 12!) / (12! 3!)
C(15, 3) = (15 × 14 × 13) / (3 × 2 × 1)
C(15, 3) = 455.
Answer: The possible number of ways = 455.

Example 2
There are 75 people at a party. Each of them shakes hand with each of the others exactly once. Then how many handshakes happen there?
Solution:
The total number of people at the party is n = 75.
For a shaking hand to happen, there must be two people.
So the number of shake hands is equal to the number of ways of selecting 2 people out of 75 people. So k = 2.
Using the combinations formula,
C (n , k) = n! / [ (nk)! k! ]
C (75, 2) = 75! / [ (75  2)! 2! ]
C (75, 2) = (75 × 74 × 73!) / (73! 2!)
C (75, 2) = (75 × 74) / (2 × 1)
C (75, 2) = 2775.
Answer: The number of handshakes = 2,775.