Combinations
Combinations refer to the selection of items from a larger set, where the order of selection does not matter. This concept is fundamental in probability and statistics, particularly in understanding the binomial distribution, which assesses the likelihood of various outcomes over multiple independent trials. For instance, one might explore the probability of obtaining a specific number of heads when tossing a coin multiple times or predicting the gender of a future child in a family with equal likelihood for boys and girls.
A useful tool for visualizing and calculating combinations is Pascal's Triangle, a mathematical arrangement where each number is the sum of the two numbers directly above it. From this triangle, one can determine the number of ways to select a certain number of items from a larger group—for example, finding out how many ways there are to choose two items from a set of seven. The number of combinations can also be represented using mathematical notation, commonly expressed as nCr, where n is the total number of items and r is the number of items to select. The symmetry of Pascal's Triangle and the availability of calculators with combination functions make exploring combinations more accessible. Understanding combinations can unlock insights in various fields, including mathematics, statistics, and everyday decision-making.
On this Page
Subject Terms
Combinations
The concept of combinations can be introduced using fairly simple examples, such as determining the probability of getting exactly three heads when a coin is tossed seven times or predicting the probability of the gender of the next child in a family of five given that boy and girl births are equally likely. On a more advanced level, the concept of combinations underpins the theory of the binomial distribution, a very important branch of statistics in its own right. Here, a given number of trials are carried out. There are two possible outcomes of each trial (success or failure). The trials are statistically independent of each other, and the probability of success is the same in every trial.
Overview
Pascal’s Triangle (see Figure 1) illustrates how combinations work. Pascal’s Triangle is named for Blaise Pascal (1623–1662), who introduced the idea of probability into statistics. Each number in a row is the sum of the two numbers immediately above it. Furthest to the right are the sums of the numbers in each row. If one tracks from the very top of the triangle down to the bottom row, the number of interest in the bottom row gives the total number of combinations. Below the triangle, the boxes are labeled slot 0, slot 1, slot 2, slot 3, slot 4, slot 5, slot 6, slot 7.
Suppose you have seven objects: A, B, C, D, E, F, G, and it is of interest to know how many ways there are of picking two of them. Simply go to the bottom row of Pascal’s Triangle in Figure 1 and look at slot 2: The triangle gives twenty-one ways of selecting two objects from the seven.
The number of ways of selecting 2 objects out of 7 is denoted by
Generalizing, the number of ways of selecting r objects from n is denoted by
Note that Pascal’s Triangle is symmetric such that
Generalizing again,
Another notation that researchers use for
is nCr. The C stands for combinations, another word for selections. Something that Blaise Pascal did not have, but what modern-day researchers do have, is calculators, which often have an nCr button.
Bibliography
Barnett, J., et al. Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science. Mathematical Assoc. of America, 2013.
Erickson, Martin. Pearls of Discrete Mathematics. Boca Raton, FL: CRC, 2009.
Fang Huaming, et al. "Binomial Distribution for Quantification of Protein Subunits in Biological Nanoassemblies and Functional Nanomachines." Nanomedicine: Nanotechnology, Biology and Medicine. 10. 7 (2014).
Roussas, George G. Introduction to Probability. 2nd ed. Oxford: Academic P, 2014.