Final answer:
There are 192 different sets of four marbles that can be made from the bag, each set containing one marble of each color red, green, yellow, and orange.
Step-by-step explanation:
To determine the number of sets of four marbles that include one of each color other than lavender, we calculate the combinations of selecting one marble of each color from their respective quantities in the bag. We have two red marbles, four green ones, four yellow marbles, and six orange marbles. To create a set with one marble of each color, we can select one red, one green, one yellow, and one orange marble.
We use the combination formula for each color: C(n, k) = n! / (k! * (n-k)!), where n is the total number of marbles and k is the number we want to choose.
For red marbles (where we choose 1 out of 2), the formula gives us C(2, 1). Similarly, for green marbles C(4, 1), for yellow marbles C(4, 1), and for orange marbles C(6, 1). We then multiply these combinations to get the total number of sets:
- C(2, 1) = 2 ways
- C(4, 1) = 4 ways
- C(4, 1) = 4 ways
- C(6, 1) = 6 ways
The total number of sets is given by the product of these combinations: 2 * 4 * 4 * 6.
Thus, there are 192 different sets of four marbles, each containing one marble of each color other than lavender.