Final answer:
The possible range of values for 'a' in the dart game score sequence, considering 46 as the median and 33 as the mode, is from 47 to 55.
Step-by-step explanation:
In the list of dart game scores 46, 25, 33, 56, 31, 46, 49, 55, a, b, and c, with the information that 33 is the mode, 46 is the median, and a > c, we can deduce the following:
As 46 is the median in a list of 11 scores, it is the 6th score when the list is ordered. Thus, there must be four scores greater than 46 which are 49, 55, a, and b assuming that b also must be greater than or equal to 46 as a > c and a and b are both greater than c.
Since 46 is not a mode, there cannot be a third instance of it among the scores a, b, or c. This implies a and b cannot be 46. Therefore, a has to be at least 47.
Now, we also know that a is greater than c, but we do not have any other scores to compare c with. Given that we don't know if c is greater than 46, 46 could potentially be c that would make a at least 47, thus the range of possible values for a starts from 47 to any number greater than that but not 56 as the next score after 55 is 56 which is not a as a > c. Therefore, a could technically be any score above 47 that is less than 56, so the range for a is 47 to 55.