What are the possible outcomes? One child can receive 4 toys, each other gets 2. Two children can receive 3 toys, the other two get 2. We will handle each case separately.

For the case that one child receives 4 toys and the other two get 2 each, choose the child who receives 4 toys. There are 4 children to choose and you choose 1 who will receive the 4 toys. Then, choose 4 of the 10 toys to give to the first child. Then, choose 2 of the remaining 6 toys to give to the second child, 2 of the remaining 4 to give to the third child, and 2 of the remaining two to give to the last child.

So, for this case, there are

$\underbrace{\binom{4}{1}}_{\begin{matrix}\text{Ch oose kid} \\ \text{to get 4 toys}\end{matrix}} \underbrace{\binom{10}{4}}_{\begin{matrix}\text{Ch oose 4 toys} \\ \text{for first child}\end{matrix}} \underbrace{\binom{6}{2}}_{\begin{matrix}\text{Cho ose 2 toys} \\ \text{for second child}\end{matrix}} \underbrace{\binom{4}{2}}_{\begin{matrix}\text{Cho ose 2 toys} \\ \text{for third child}\end{matrix}} \underbrace{\binom{2}{2}}_{\begin{matrix}\text{Cho ose 2 toys} \\ \text{for last child}\end{matrix}}$

This can be simplified as:

$\dfrac{4!\cdot 10!}{(1!3!)(4!2!2!2!)} = 75,600$

Next, choose two children to get 3 toys. Then choose 3 of the 10 toys for the first child, 3 of the remaining 7 for the second child, 2 of the remaining 4 for the third, and 2 of the remaining 2 for the last:

$\underbrace{\binom{4}{2}}_{\begin{matrix}\text{Ch oose kids} \\ \text{to get 3 toys}\end{matrix}} \underbrace{\binom{10}{3}}_{\begin{matrix}\text{Ch oose 3 toys} \\ \text{for first child}\end{matrix}} \underbrace{\binom{7}{3}}_{\begin{matrix}\text{Cho ose 3 toys} \\ \text{for second child}\end{matrix}} \underbrace{\binom{4}{2}}_{\begin{matrix}\text{Cho ose 2 toys} \\ \text{for third child}\end{matrix}} \underbrace{\binom{2}{2}}_{\begin{matrix}\text{Cho ose 2 toys} \\ \text{for last child}\end{matrix}}$

This can be simplified as:

$\dfrac{4!\cdot 10!}{(2!2!)(3!3!2!2!)} = 151,200$

Adding these together: $75,600+151,200 = 226,800$