The number of unordered samplings of k elements from a collection of n elements with replacement is (n+k-1)Ck.
To determine the number of unordered samplings of k elements from a collection of n elements with replacement, consider the following approach:
Representing the sampling as a multiset:
An unordered sampling of k elements from a collection of n elements with replacement can be represented as a multiset, which is a collection of elements where each element may appear multiple times. In this case, the multiset would consist of k elements, each representing one of the sampled elements.
Mapping the multiset to a partition:
Each unique multiset corresponds to a unique partition of the positive integer k. A partition of k is a set of positive integers that sum to k, where the order of the integers does not matter. For example, the partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, and {1, 1, 1, 1}.
Counting the partitions of k:
The number of partitions of k is given by the formula (n+k-1)Ck, which is also known as the number of ways to choose k objects from a set of n+k-1 objects, where order does not matter and repetition is allowed.
Therefore, the number of unordered samplings of k elements from a collection of n elements with replacement is equal to (n+k-1)Ck.