Final answer:
To distribute 12 pieces of candy to 4 kids, taking into account that two of the kids are twins and must receive an equal number of candies, there are 18304 ways.
Step-by-step explanation:
To distribute 12 pieces of candy to 4 kids, taking into account that two of the kids are twins and must receive an equal number of candies, we can use the concept of combinations. We can distribute the candies among the non-twin kids in 13C3 ways, which is equal to 286.
Then, we need to divide the remaining candies between the twin kids. We can distribute the remaining 6 candies between the twins in 7C0 ways for both twins receiving no candies, plus 6C1 ways for one twin receiving 1 candy and the other receiving 5 candies, plus 6C2 ways for one twin receiving 2 candies and the other receiving 4 candies, and so on up to 6C6 ways for both twins receiving 6 candies each.
Adding up all these possibilities, we get the final answer.
The total number of ways to distribute 12 pieces of candy to 4 kids, taking into account that two of the kids are twins and must receive an equal number of candies, is 286 * (1 + 6 + 15 + 20 + 15 + 6 + 1) = 286 * 64 = 18304 ways.