Hey Developer’s, I’m back with a new topic which is Binomial Coefficients in the series of statistics foundations.

### Quick Refresher

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

C(n,k) = P(n,k)/k! = n!/k!(n-k)!

So, let’s get started …

### Binomial Coefficients

A binomial coefficient equals the number of combinations of k items that can be selected from a set of n items.

If your observations are independent, each represents one of two outcomes (think: success and failure), your number of trials is 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

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 p^{r}q^{n-r}.

**Binomial Coefficient Examples**

Example : A class has 15 girls and 30 boys. Pick 10 children at random. What’s the probability you’ll pick exactly 3 girls ?

Number of ways of picking 3 girls from 15 girls is : 15**C**3

Number of ways of picking 7 boys from 30 is : 30**C**7

Overall number of combinations is : 45**C**10

Therefore, P(3 girls) = 15**C**3 * 30**C**7/45**C**10 = 0.29

#### Next Post will be on Multinomial Coefficients

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