To find the number of ways in which each child can choose a flavor of ice cream such that exactly three children choose the same flavor, we can break down the problem into cases.
Case 1: Three children choose the same flavor, and the other three children choose different flavors. In this case, we need to select one flavor for the three children who choose the same flavor. There are 3 ways to do that. Then, we need to select three different flavors for the remaining three children, which can be done in 3! = 6 ways. Therefore, the total number of ways for this case is 3 * 6 = 18.
Case 2: Four children choose one flavor, and the other two children choose a different flavor. In this case, we need to select one flavor for the four children who choose the same flavor, which can be done in 3 ways. Then, we need to select one flavor for the two remaining children, which can be done in 2 ways. Therefore, the total number of ways for this case is 3 * 2 = 6.
Case 3: Five children choose one flavor, and the remaining child chooses a different flavor. In this case, we need to select one flavor for the five children who choose the same flavor, which can be done in 3 ways. Then, we need to select one flavor for the remaining child, which can be done in 2 ways. Therefore, the total number of ways for this case is 3 * 2 = 6.
Case 4: All six children choose the same flavor. In this case, we need to select one flavor for all six children, which can be done in 3 ways. Therefore, the total number of ways for this case is 3.
Finally, we sum up the results from all cases to get the total number of ways: 18 + 6 + 6 + 3 = 33.
Therefore, there are 33 ways in which each child can choose a flavor of ice cream so that exactly three children choose the same flavor.
