Where do you get $2*6!$ for (a)? I find $7!$
For the three week problem, let us start by assuming $3,2,2$. We can multiply by $3$ at the end to take care of cyclic permutations of weeks. The two pairs are differently named bins in this case. The two C's can be together in one of the twos in $2\text{(which week)}*{5 \choose 3}\text{(who is in the triplet)}=20$ ways. They can be two of the triplet in $5\text{(the other member of the triplet)}*{4 \choose 2}\text{(the first other couple)}=30$ ways. Multiplying by $3$ gives a total of $150$ ways.
You have overcounted the number of ways to distribute the toys. Suppose the toys are $T_1,T_2,T_3,T_4$. Then one of the toy distributions you counted is where you pick out $T_1,T_2,T_3$ and distribute them to person 1, person 2, and person 3, in that order, and give $T_4$ to person 1. But you've also counted, as a separate way of distributing the toys, the distribution where you pick out $T_2,T_3,T_4$ and give $T_2$ to person 2, $T_4$ to person 3, and $T_4$ to person 1, and then give $T_1$ to person 1. This is, however, the same toy distribution as the previous one.
You can count the number of ways to distribute the toys as follows. Exactly one person must receive two toys, and there are $3$ possibilities for who that person is. There are ${4 \choose 2}=6$ choices for which toys that person receives. There are two remaining toys and two people who must each receive one of them, so there are $2$ ways to distribute the last two toys. This gives $3 \cdot 6 \cdot 2 = 36$ ways to distribute the toys.
Now you can multiply as you did before to get $6 \cdot 36=216$ ways to distribute the marbles and toys.
Best Answer
We can do it by splitting into cases: $7$-$1$-$1$, $5$-$3$-$1$, $3$-$3$-$3$.
First case: The lucky person who gets $7$ pies can be chosen in $\binom{3}{1}$ ways. Her pies can be chosen in $\binom{9}{7}$ ways. For each choice, the older of the unlucky people can be assigned her pie in $\binom{2}{1}$ ways, for a total of $\binom{3}{1}\binom{9}{7}\binom{2}{1}$.
Next two cases: it's your turn. Then add up.