Binomial Coefficient Calculator
Last updated: August 24, 2022
This Binomial Coefficient Calculator will calculate the value of the Binomial Coefficient.
What is a Binomial Coefficient?
The binomial coefficient (n; k) is the number of ways of picking k unordered outcomes from n possibilities.
There are various ways to write the notation of binomial coefficients:
{\dbinom{n}{k}} \; = \; C{\dbinom{n}{k}} \; = \; {_nC_k} \; = \; C_{n,k} = C(n,k)
The Binomial Coefficient is also called “n choose k”.
Example: Let’s say there are 5 balls of different colors in the basket – blue, green, yellow, red and white.
How many ways are there to get 2 balls from the basket?
Let’s write down all the ways:
- blue and green,
- blue and yellow,
- blue and red,
- blue and white,
- green and yellow,
- green and red,
- green and white,
- yellow and red,
- yellow and white,
- red and white.
We got 10 ways. This is the answer.
But the answer can be obtained without brute force. Just use the Binomial Coefficient Formula.
Binomial Coefficient Formula
where is the symbol ! – factorial.
Let’s go back to the problem and insert the values into the formula:
{\dbinom{5}{2} = \dfrac{5!}{2! \cdot (5 – 2)!} = \dfrac{5!}{2! \cdot 3!} = \dfrac{120}{2 \cdot 6} = \dfrac{120}{12} = 10}
This problem (n choose k) can be solved using a calculator – 5 choose 2.
Examples
Question: How many ways can you give away 32 playing cards to 6 player?
Solution:
From the question, it is clear that:
n = 32
k = 6
Use the formula:
{\dbinom{32}{6} = \dfrac{32!}{6! \cdot (32 – 6)!} = \dfrac{32!}{6! \cdot 26!} = 906192}
Question: How many different quartet can be selected from group 25 students?
Solution:
From the question, it is clear that:
n = 25
k = 4
Use the formula:
{\dbinom{25}{4} = \dfrac{25!}{4! \cdot (25 – 4)!} = \dfrac{25!}{4! \cdot 21!} = 12650}
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