Understanding permutations, combinations, and solving counting problems.
Combinatorics is the branch of mathematics concerned with counting, arrangement, and combination of objects. In computer science, it is essential for analyzing algorithms, calculating probabilities, and solving a wide class of problems that involve enumeration. The two most fundamental concepts are permutations and combinations. A permutation is an arrangement of objects in a specific order. The number of permutations of 'n' distinct objects is n! (n factorial). The number of ways to arrange 'k' objects chosen from 'n' distinct objects is given by P(n, k) = n! / (n-k)!. A combination, on the other hand, is a selection of objects where the order does not matter. The number of ways to choose 'k' objects from a set of 'n' distinct objects is given by the binomial coefficient C(n, k) = n! / (k! * (n-k)!). Problems often require calculating these values, which can involve large factorials. When working with modular arithmetic, we need to compute nCr % m. This requires precomputing factorials and their modular inverses. Other important principles include the pigeonhole principle, which states that if 'n' items are put into 'm' containers, with n > m, then at least one container must contain more than one item. This simple idea can be used to prove surprisingly complex results.