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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.

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

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