 # n Choose k Formula

The formula is known as n Choose k, as from the name, it allows us to choose k elements from n elements. To be more precise, by this formula we can calculate the number of ways of choosing k elements. It is denoted by $${n \choose k}$$.

The notation was introduced by Andreas von Ettingshausen in 1826.

Index

## The n Choose kFormula

Coming to the formula of n choose k:

$${n \choose k} = \frac{n!}{k!(n – k)!}$$

Where, $$n \geq k \geq 0$$

The formula is also famously known as the binomial coefficient. Binomial coefficients are used in many areas of mathematics and especially in combinatorics.

Alternative notations to this include $$C(n, k), _{n}C_k, ^{n}C_k, C^n_k, C^k_n \mbox{ and } C_{n, k}$$.

## Computation of Binomial Coefficients

Binomial coefficients can be computed in several ways, without having to expand the formula:

### Recursive Formula

$${n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}$$

for all integers $$n, k$$ such that $$1 \leq k \leq n-1$$.

### Multiplicative Formula

$${n \choose k} = \frac{n(n-1)(n-2) … (n-(k + 1))}{k(k-1)(k-2)(k-3) … 1}$$

### Factorial Formula

$${n \choose k} = \frac{n!}{k!(n – k)!}$$

## Applications

• The binomial coefficient is highly used in combinatorics.
• It also finds applications in statistics specifically in the concept of the binomial distribution.
• It is also used in binomial theorem, in elementary algebra.

## Solved Examples

Question 1. Find the number of ways of forming a cricket team of 11 players out of 20 players.

Solution. Given, n = 20, k = 11

As we know,

$${n \choose k} = \frac{n!}{k!(n – k)!}$$

⇒ $${20 \choose 11} = \frac{20!}{11!(20 – 11)!}$$

⇒ $${20 \choose 11} = \frac{20*19*18*17*16*15*14*13*12}{9*8*7*6*5*4*3*2*1}$$

⇒ $${20 \choose 11} = 167960$$

Hence, there are 167960 ways of choosing 11 players out of 20 players to form a cricket team.

Question 2. How many possible ways are there to draw 5 cards out of a set of 10 cards?

Solution. Given, n = 10, k = 5

As we know,

$${n \choose k} = \frac{n!}{k!(n – k)!}$$

⇒ $${10 \choose 5} = \frac{10!}{5!(10 – 5)!}$$

⇒ $${10 \choose 5} = \frac{10*9*8*7*6}{5*4*3*2*1}$$

⇒ $${10 \choose 5} = 252$$

Thus, there are 252 possible ways to choose 5 cards out of 10 cards.

## FAQs

How to calculate n choose k?

n choose k, also popularly known as binomial coefficient can be calculated using the formula,
$${n \choose k} = \frac{n!}{k!(n – k)!}$$

Is n choose k combination or permutation?

The binomial coefficient $${n \choose k}$$ essentially comes under combination. The formula computes the different ways in which different combinations of k items can be chosen out of n items.
Another denotation for n choose k is C(n, k), where C essentially stands for combinations.

What does n choose k equal?

The binomial coefficient $${n \choose k}$$ is used to find the number of ways in which k items can be chosen out of n items, given that n ≥ k ≥ 0.
It can be calculated using the formula, $${n \choose k} = \frac{n!}{k!(n – k)!}$$

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