Arrangements and Combinations · Arrangements with Repetitions

Lesson 8

Nikolai Chukhin · Alexander S. Kulikov

As we already know, the binomial coefficient \(\binom{n}{k}\) is:

  • the coefficient of \(a^{n-k}b^{k}\) in \((a+b)^{n}\);

  • the number of ways to split a set of size \(n\) into two subsets, the first of which has size \(k\), and the second — \((n-k)\);

  • the number of words of length \(n\) in the alphabet \(\{1, 2\}\), in which \(k\) ones and \(n-k\) twos appear.

Looking at these definitions, it is natural to ask: what happens if we consider the \(n\)th power of a sum not of two variables but, say, \(t\) variables? What if we split sets into \(t\) parts? What if strings are in an alphabet of size \(t\)? And indeed, such a generalization exists, and the result is the same number.