The equation C(n, k-1) + nC(n, k) = C(n + 1, k) is known as the Pascal's rule or Pascal's identity, which relates the binomial coefficients C(n, k) and C(n + 1, k).
To understand how the equation works, we can break down each term:
- C(n, k-1) represents the number of ways to choose k-1 objects from a set of n objects.
- nC(n, k) represents the number of ways to choose k objects from a set of n objects, multiplied by the number of ways to choose one additional object from the remaining n-k objects.
- C(n + 1, k) represents the number of ways to choose k objects from a set of n+1 objects.
By combining these terms, we can see that the left-hand side of the equation represents the total number of ways to choose k objects from n+1 objects. This is equivalent to the right-hand side of the equation, which also represents the number of ways to choose k objects from n+1 objects.
Therefore, Pascal's identity can be used to simplify calculations involving binomial coefficients, making it a powerful tool in combinatorics and probability theory.