The formula is known as ** n Choose k**, as from the name, it allows us to

**choose**. To be more precise, by this formula we can calculate the

*k*elements from*n*elements*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 k** **Formula**

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