Final answer:
The number of ways eight children can be selected from a group of 15, ensuring Danylo and Shuying are not both included, is represented by 2C1 x 13C7, which accounts for two separate scenarios where one is included and the other is not.
Step-by-step explanation:
To find out the number of ways eight children can be selected from a group of 15, where Danylo and Shuying must not both be included, we need to consider two scenarios: one where Danylo is included but Shuying is not, and vice versa. Since there are 15 children in total, if we include Danylo, we are left with 14 children (excluding Shuying), from which we need to select 7 more children to make a total of 8. This can be done in 13C7 ways according to the combination formula. Similarly, if Shuying is included and Danylo is not, we again have 13C7 ways of choosing the remaining 7 children. Since both scenarios are mutually exclusive, we add them together. Therefore, the answer is 13C7 + 13C7, which simplifies to 2 x 13C7. Option (a) 2C1 x 13C7 is the correct answer, representing the sum of the two scenarios.