# Combination Formula

"Combination" in math means "selection". The combination formula is used to find the number of ways to select 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 any 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.

### Combination Formula

The combination formula is,

C (n , k) (or) \(n_{ C_{k}}\) (or) \( _{n} C_{k}\) (or) \(\left(\begin{array}{l}n \\k\end{array}\right)\) = \(\dfrac{n!}{(n-k)! k!}\)

where

- n = total number of things
- k = number of things that are chosen (or) selected

## Combination Formula Derivation

We know that the number of ways of selecting 'k' different things from 'n' different things is the combination which is denoted by C (n, k). Also, we know that the number of ways of selecting and arranging 'k' different things from 'n' different things is the permutation which is denoted by P (n, k). i.e.,

Permutation = combination AND arrangement

We know that r different things can be arranged among themselves in k! ways. Then by using the above statement and the fundamental counting principle,

P(n, k) = C(n, k) × k!

Using the permutation formula, P(n, k) = n! / [ (n-k)! ]. Substituting this in the above equation,

n! / [ (n-k)! ] = C(n, k) × k!

C(n, k) = n! / [ (n-k)! k! ]

Thus, the combination formula is derived.

## 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! / [ (n-k)! 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 hands 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! / [ (n-k)! 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.

**Example 3: **In a group of 12 people, 3 prizes, each worth $10, are distributed. In how many ways this can be done?

**Solution:**

The total number of people is n = 12.

The number of prizes is r = 3.

Since all the prizes worth the same, the arrangement of prizes doesn't matter, it is only the selection.

So we use the combination formula to find the required number of ways.

C (n , k) = n! / [ (n-k)! k! ]

C (12, 3) = 12! / [ (12 - 3)! 3! ]

= (12 × 11 × 10 × 9!) / (9! 3!)

= (12 × 11 × 10) / (3 × 2 × 1)

= 220

**Answer: **The requied number of ways = 220.

## FAQs on Combination Formula

### What Is the Combination Formula?

The combination formula is a formula that is used to find the number of ways of selecting 'k' things out of 'n' things, which is usually denoted by C (n , k). The combination formula is, C (n , k) = n! / [ (n-k)! k! ].

### How To Derive the Combination Formula?

Let us consider that there are 'n' number of things out of which 'k' things are selected (and/or) arranged. Then, P(n, k) = n! / [ (n-k)! ] gives the number of ways of selecting and arranging k things out of n things. If C (n, k) is the number of ways of only selecting n things out of n things, then by the fundamental principle of counting

n! / [ (n-k)! ] = C(n, k) × k! (as k things can be arranged in k! ways among themselves)

C(n, k) = n! / [ (n-k)! k! ]

### What Are the Applications of Combination Formula?

The combination formula is used to find the number of ways of selecting k things out of n things when there is no importance for the ordering. This formula is helpful in many real-life situations such as finding the number of teams that can be formed from a group of people, the number of ways in which some prizes (of the same worth) can be distributed among some students, etc.

### How To Identify When to Use the Combination Formula?

If the problem involves only selection and there is no importance for arrangement, then we have to use the combination formula. For example, if we have to find the number of ways of selecting 3 eggs from 12 eggs, we have to use the combination formula as it is just selection (and no arrangement). On the other hand, if we have to find the number of words that can be formed by using 5 alphabets, it is NOT the combination formula (in fact, we have to use the permutation formula here) as forming a word involves arrangement too.