Counting Unordered Sequences with Repetition

From a set A containing numbers 1 through 6, determine how many sequences of three objects can be selected with repetition and unordered.

This problem focuses on the combinatorial method of counting sequences where order does not matter and repetition is allowed. Such problems can often be solved using the stars and bars approach, a popular technique in combinatorics for determining the number of ways to place indistinguishable objects into distinct boxes. In this problem, think about the 'objects' as the sequences you form and the 'boxes' as the numbers 1 through 6 in set A.

When repetition is allowed and order doesn't matter, this is equivalent to finding the number of combinations with repetition, also known as multisets. Conceptually, you're choosing how many times each number from 1 to 6 can appear in a sequence of length three, where the sequence doesn't care about the arrangement of the chosen numbers. Understanding the difference between combinations (where order doesn't matter and items are indistinguishable) and permutations (where order does matter) is key in problems of this nature.

This problem serves as a practice in basic combinatorial techniques which are fundamental in discrete mathematics and are applicable in various fields such as computer science, probability theory, and algorithm design. The challenge is to apply the combinatorial logic and possibly derive a formula to calculate the number of such sequences efficiently.