As you correctly said, we can immediately give 1 peach, 1 apple and 1 pineapple to each of the 3 kids, and now we have 7 peaches, 1 apple and 3 pineapples.
We can distribute the remaining apple in 3 ways: give it to the first kid, to the second or to the third one.
In how many ways can we distribute indistinguishable pieces of fruit to kids, which are of course distinguishable? It can be shown that we can do it in ways.
So, we can distribute 7 remaining peaches to 3 kids in ways, and we can distribute 3 remaining peaches to 3 kids in ways.
To find the total number of possible distributions, we multiply the above numbers to get that the final answer is .