A very nice puzzle and many people sent me answers. I was disappointed that not too many children tried it. Even though I am posting the solution, I would encourage you to try to puzzle with your children. Many people got it right, that included Suman Saraf, Abhinav Jain, Karan Sharma, Vishal Poddar and Aman Goyal.
The answer is 1, 1, 7. Here is the answer explained through a simple video for children:
For folks who prefer to just read, here is the explanation:
The possible outcomes are 009, 018, 027, 036, 045, 117, 126, 135, 144, 225, 234, 333. Belle was not able to answer, so the number is not 333. Carol was not able to guess so the number is not 009. Nick was not able to guess, so the number is not 018. After Nick was not able to answer the number Carol was able to find the number, this implies that there were 2 possibilities from the second number. As Carol was able to find the number, this means that the second digit is 1 and if it is not 018, then the number is 117.
Hope you all enjoyed the puzzle!