In how many ways can we distribute 14 identical candies among three children so that each child receives at least three candies?
First, we need to determine the number of ways to obtain 14 as the sum of three integer parts, each of which is greater than or equal to 3, that is,

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-077.jpg?height=108&width=466&top_left_y=2056&top_left_x=818)

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-077.jpg?height=274&width=962&top_left_y=2210&top_left_x=290)

Now, for each of these possibilities, we can make different distributions among the three children, as shown in the following table. Note that when the three parts are different, we have six possibilities, and when two are the same, we have only three possibilities.

|  | 1st child | 2nd child | 3rd child |
| :---: | :---: | :---: | :---: |
| $14=3+3+8$ | 3 | 3 | 8 |
|  | 3 | 8 | 3 |
|  | 8 | 3 | 3 |
| $14=3+4+7$ | 3 | 4 | 7 |
|  | 3 | 7 | 4 |
|  | 4 | 3 | 7 |
|  | 4 | 7 | 3 |
|  | 7 | 3 | 4 |
|  | 7 | 4 | 3 |
| $14=3+5+6$ | 3 | 5 | 6 |
|  | 3 | 6 | 5 |
|  | 5 | 3 | 6 |
|  | 5 | 6 | 3 |
|  | 6 | 3 | 5 |
|  | 6 | 5 | 3 |
| $14=4+4+6$ | 4 | 4 | 6 |
|  | 4 | 6 | 4 |
|  | 6 | 4 | 4 |
| $14=4+5+5$ | 4 | 5 | 5 |
|  | 5 | 4 | 5 |
|  | 5 | 5 | 4 |

Thus, we have $3+6+6+3+3=21$ different ways to distribute the candies among the three children.