The formula to calculate the Stirling Number of the Second Kind (S(n, k)) is:
\[ S(n, k) = k \times S(n - 1, k) + S(n - 1, k - 1) \]
Where:
Let's say the total number of objects (\( n \)) is 5 and the number of objects in each partition (\( k \)) is 3. Using the formula:
\[ S(5, 3) = 3 \times S(4, 3) + S(4, 2) \]
We get:
\[ S(5, 3) = 3 \times 6 + 7 = 25 \]
So, the Stirling Number of the Second Kind (\( S(5, 3) \)) is 25.
The Stirling numbers of the second kind, denoted as \( S(n, k) \), represent the number of ways to partition a set of \( n \) objects into \( k \) non-empty subsets. They are used in various fields of mathematics, including combinatorics, algebra, and probability theory. The Stirling numbers have applications in solving problems related to partitions of sets, counting functions, and the distribution of objects into indistinguishable boxes.
