Probability theory quantifies uncertainty, and counting is its foundation. Before you can compute the probability of an event, you need to know how many outcomes are possible and how many of those outcomes satisfy your event. Combinatorics, the mathematics of counting, provides the tools to answer these questions systematically.



The multiplication principle states that if one task has m ways and a second independent task has n ways, the combined task has m x n ways. This simple idea underlies all of combinatorics. With it, you can count the number of possible license plates, passwords, or DNA sequences.



Permutations count arrangements where order matters. The number of ways to arrange r items from n distinct items is nPr = n! / (n-r)!. The number of ways to arrange all n items is n!. For example, a deck of 52 cards can be arranged in 52! ways, which is about 8 x 10^67, vastly more than the number of atoms on Earth. Our permutation and combination calculator computes these values along with step-by-step explanations.



Combinations count selections where order does not matter. The number of ways to choose r items from n is nCr = n! / (r! x (n-r)!), also written as C(n, r) or the binomial coefficient. The difference from permutations is the division by r!, which accounts for all the orderings of the same selection that we no longer distinguish.



Pascal’s triangle provides a beautiful visual representation of binomial coefficients. Each entry is the sum of the two entries above it, reflecting the identity C(n, r) = C(n-1, r-1) + C(n-1, r). The binomial theorem, (a+b)^n = sum C(n,r) a^r b^(n-r), connects these coefficients to polynomial expansion. Our binomial expansion calculator generates these expansions automatically.



In probability, the formula P(event) = favorable outcomes / total outcomes assumes equally likely outcomes. When outcomes are not equally likely, you count using combinatorics and weight by probability. The factorial calculator helps with large factorial computations needed for probability problems.



Conditional probability and Bayes’ theorem extend basic probability to situations where you have prior information. P(A|B) = P(B|A)P(A) / P(B). This formula is used in medical testing (what does a positive test result really mean?), spam filtering, and machine learning.



The binomial distribution models the number of successes in n independent trials with success probability p. Its probability mass function uses binomial coefficients: P(X = k) = C(n, k) p^k (1-p)^(n-k). The normal distribution approximates the binomial for large n. The Poisson distribution models rare events and is derived from the binomial in the limit as n approaches infinity and p approaches zero.



Mastering combinatorics requires practice with identifying whether order matters (permutation vs combination), whether items are distinct or identical, and whether selections are made with or without replacement. The calculator tools on this site let you verify your reasoning and build intuition through experimentation.