Binomial Coefficient

Statistics Definitions >

Binomial coefficients tell us how many ways there are to choose k things out of larger set. More formally, they are defined as the coefficients for each term in (1+x)n. Written as binomial coefficient, (read n choose k), where binomial coefficient is the binomial coefficient of the xk term of the polynomial.

An alternate notation is nCk.

For non-negative integer values of n (number in the set) and k (number of items you choose), every binomial coefficient nCk is given by the formula:
Binomial Coefficient - non-negative integer values formula
The “!” symbol is a factorial.


Imagine you have 5 elements {a, b, c, d, f}. To find out how many different subsets of 2 elements it has, look at the binomial coefficient 5C2.
Binomial Coefficient - example formula with 5 elements
This equals 10.

For a more concrete example, suppose the president of a student club must pick three members of an advisory board from a faculty pool of 24. To find out how many ways he could make this choice, look at the 24C3 binomial coefficient, 24!/(3! 21!)= 2024. So the president of the student club has 2024 cabinet choices.

Importance of the Binomial Coefficient in Statistics

The binomial coefficient is much more than just a simple formula to calculate how many ways you can pull an advisory board from a candidate pool, a 4-digit pin from a 10-digit set, or a plate of apples from a bin of the same. It’s also part of the description of the binomial distribution, a simple probability distribution for frequently encountered 2-outcome situations.

If your observations are independent, each represents one of two outcomes (think: success and failure), your number of trials are fixed and the probability of success is the same for each trial, then the probability you have exactly r successes during your n independent trials will be
Binomial distribution formula
This formula represents the binomial distribution. Here p is the probability of success in each instance, and q=1-p, the probability of failure.

The binomial coefficient n choose r tells you how many success-failure sequences, of the set of all possible sequences, will result in exactly r successes.  The probability of each of those individual sequences happening is just  prqn-r.


Binomial Coefficients @ Dartmouth. Retrieved September 27, 2017 from:
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