a bag contains 3 red marbles, 5 blue marbles, 2 yellow marbles, and 2 lavender marbles. How many sets of 4 marbles includes at most one of the yellow marbles?

To solve this problem, we need to consider two scenarios: one where we have zero yellow marbles in the set, and another where we have exactly one yellow marble in the set. Then, we sum up the number of sets from each scenario to get the final answer.

Scenario 1: No yellow marbles in the set
In this case, we need to select 4 marbles from the remaining 12 (since we are not considering the yellow marbles). We need to calculate the number of ways to choose 4 marbles from a set of 12:

C(12, 4) = 12! / (4!(12 - 4)!) = 495

Scenario 2: Exactly one yellow marble in the set
We have two yellow marbles, so we have two choices for selecting one yellow marble and then we need to select 3 marbles from the remaining 12 (since we are considering only one yellow marble). We need to calculate the number of ways to choose 3 marbles from a set of 12:

C(12, 3) = 12! / (3!(12 - 3)!) = 220

Finally, we add up the number of sets from both scenarios to get the total number of sets that include at most one yellow marble:

495 + 220 = 715

Therefore, there are 715 sets of 4 marbles that include at most one of the yellow marbles.