Final answer:
Using combinatorics and the stars and bars method, part a has (17 choose 2) ways to distribute the marbles, and part b has (11 choose 2) ways to distribute the marbles ensuring each jar has at least two marbles.
Step-by-step explanation:
The question is about determining the number of ways to distribute marbles into jars.
For part a, if each marble is the same, we use the stars and bars method. We have 15 stars (marbles) and 2 bars (divisions between jars), and we need to arrange those in a row. This can be done in combinatorial ways calculated by the formula (n+k-1 choose k-1), where n is the number of marbles and k is the number of jars. Therefore, there are (15+3-1 choose 3-1) = (17 choose 2) ways.
For part b, since each jar must have at least two marbles, we first place two marbles in each jar, leaving 15 - 3*2 = 9 marbles to be distributed freely. Again using the stars and bars method, we have 9 stars and 2 bars, yielding (9+3-1 choose 3-1) = (11 choose 2) arrangements.
In combinatorics, problems involving the distribution of identical items into distinct categories can often be solved using the stars and bars technique, which is an application of the combinatorial counting principle.