Answer:
a. If each marble is different, there are 3 jars and 15 marbles, so we need to calculate 3^15 = 14,348,907 possible ways.
b. If each marble is the same, we can use stars and bars to count the number of ways to distribute the marbles among the jars. We need to divide 15 marbles among 3 jars, so we can place 2 dividers (representing the boundaries between the jars) among 14 objects (15 marbles minus 1), and there are C(14,2) = 91 ways to do this.
c. If each marble is the same and each jar must have at least two marbles, we can first distribute 6 marbles among the jars (using the method from part b, with 2 stars and 2 bars), and then distribute the remaining 9 marbles among the jars (again using the same method, but with 1 star and 2 bars). This gives C(6+3-1,2) * C(9+3-1,2) = C(8,2) * C(11,2) = 28 * 55 = 1,540 possible ways.
d. If each marble is the same but each jar can have at most 6 marbles, we can use generating functions to count the number of ways to distribute the marbles. The generating function for each jar is (1 + x + x^2 + ... + x^6), and the generating function for all three jars is (1 + x + x^2 + ... + x^6)^3. The desired coefficient of the x^15 term can be found using the multinomial coefficient C(15+3-1, 3-1, 3-1, 3-1) = C(17,2,2,2) = 15,015. Therefore, there are 15,015 possible ways.
e. If you have 10 identical red marbles and 5 identical blue marbles, we can again use stars and bars to distribute the marbles among the jars. We need to distribute 10 red marbles and 5 blue marbles among 3 jars, so we can place 2 dividers among 14 objects (10 red marbles, 5 blue marbles, and 2 dividers), which gives C(14,2) = 91 possible ways.
Step-by-step explanation: