In a typical \(6/49\) lotto, where \(6\) numbers are drawn from a range of \(49\) and if the six numbers drawn match on the ticket, the ticket holder is a jackpot winner. The odds of this event are \(1\) in \(13,983,816\).
The study of counting methods dates at least to Gottfried Wilhelm Leibniz’s Dissertatio de Arte Combinatoria in the seventeenth century. Combinatorics has applications in many fields of study. Applications of combinatorics arise, for example in chemistry, in studying the arrangements of atoms in molecules and crystals; biology in questions about the structure of genes and proteins; physics, in problems in statistical mechanics; communication, in the design of codes for encryption, compression, and especially in computer science, for instance in problems of scheduling and allocating resources.
The links to my notes on how to count and the solved problems are given below.