There is a better approach. Let $x_k$ be the number of apples received by the $k$th child. Then
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 10$$
is an equation in the positive integers. A particular solution corresponds to the placement of five addition signs in the nine spaces between successive ones in a row of ten ones.
$$1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1$$
For instance,
$$1 1 + 1 + 1 1 1 + 1 + 1 + 1 1$$
corresponds to the solution $x_1 = 2$, $x_2 = 1$, $x_3 = 3$, $x_4 = x_5 = 1$, $x_6 = 2$. The number of such solutions is the number of ways we can select five of the nine spaces in which to place an addition sign, which is
$$\binom{9}{5}$$
Addendum: Using Barry Cipra's observation, we can confirm this result by using your method.
One child receives five apples and the other five children each receive one apple: There are $6$ ways to select the child who receives five apples.
One child receives four apples, another child receives two apples, and each of the other four children each receive one apple: There are six ways to choose the child who receives four apples and five ways to choose the child who receives two apples. Hence, there are $6 \cdot 5 = 30$ such distributions.
Two children each receive three apples and the other four children each receive one apple: There are $$\binom{6}{2} = 15$$ ways to select the two children who each receive two apples.
One child receives three apples, two children each receive two apples, and the other three children each receive one apple: There are six ways to choose the child who receives three apples and $\binom{5}{2}$ ways to choose which two of the other five children each receive two apples. Hence, there are $$\binom{6}{1}\binom{5}{2} = 6 \cdot 10 = 60$$ such distributions.
Four children each receive two apples and the other two children receive one apple: There are
$$\binom{6}{4} = 15$$
ways to select which two children will receive two apples.
Observe that
$$6 + 30 + 15 + 60 + 15 = \binom{9}{5}$$
Your answers to the first two questions are correct.
In how many ways can $15$ indistinguishable candies be distributed to five children if child $C$ and child $D$ receive $7$ candies together?
We must distribute seven candies among the children $C$ and $D$ and eight candies among the children $A$, $B$, and $E$. The number of ways we can distribute the candies to children $C$ and $D$ is eight since $C$ must receive between $0$ and $7$ candies inclusive, with $D$ receiving the rest. The number of ways the remaining eight candies can be distributed to the children $A$, $B$, and $D$ is
$$\binom{8 + 3 - 1}{3 - 1} = \binom{10}{2}$$
as you correctly found. Hence, the number of ways of distributing to the five children if $C$ and $D$ receive exactly seven candies between them is
$$\binom{7 + 2 - 1}{2 - 1}\binom{8 + 3 - 1}{3 - 1} = \binom{8}{1}\binom{10}{2}$$
In how many ways can $15$ indistinguishable candies be distributed to five children if no child receives more than six candies?
Let $x_i$, $1 \leq i \leq 5$, be the number of candies received by the $i$th child. Then we seek the number of solutions of the equation
$$x_1 + x_2 + x_3 + x_4 + x_5 = 15 \tag{1}$$
in the nonnegative integers subject to the restrictions that $x_i \leq 6$ for $1 \leq i \leq 5$.
A particular of equation 1 corresponds to the placement of four addition signs in a row of $15$ ones. For instance,
$$1 1 1 + 1 1 1 1 + 1 1 + + 1 1 1 1 1 1$$
corresponds to the solution $x_1 = 3$, $x_2 = 4$, $x_3 = 2$, $x_4 = 0$, and $x_5 = 6$. The number of such solutions is the number of ways we can place four addition signs in a row of fifteen ones, which is
$$\binom{15 + 5 - 1}{5 - 1} = \binom{19}{4}$$
since we must choose which four of the nineteen positions required for fifteen ones and four addition signs will be filled with addition signs.
By similar reasoning, the number of solutions of the equation
$$x_1 + x_2 + x_3 + \cdots + x_n = k$$
in the nonnegative integers is
$$\binom{k + n - 1}{n - 1}$$
since we must choose which $n - 1$ of the $k + n - 1$ positions required for $k$ ones and $n - 1$ addition signs will be filled with addition signs.
From these, we must subtract those cases in which at least one child receives more than six candies. Observe that at most two children could receive more than six candies since $2 \cdot 7 = 14 < 15 < 21 = 3 \cdot 7$.
Suppose a child receives more than six candies. There are five ways to choose that child. We give that child seven candies. The remaining eight candies can be distributed among the five children in
$$\binom{8 + 5 - 1}{5 - 1} = \binom{12}{4}$$
ways. Hence, there are
$$\binom{5}{1}\binom{12}{4}$$
ways to distribute the candies in such a way that a child receives more than six candies.
However, if we subtract this amount from the total, we will have subtracted too much since we have counted each case in which two children receive more than six candies twice, once for each way of designating one of those children as the child who received more than six candies. We only want to subtract those cases once, so we must add those cases back.
Suppose two children each receive more than six candies. There are $\binom{5}{2}$ ways to select those two children. Give each of them seven candies. That leaves one candy to distribute among the five children, which can be done in five ways. Hence, the number of distributions in which two children receive more than six candies is
$$\binom{5}{2}\binom{1 + 5 - 1}{5 - 1} = \binom{5}{2}\binom{5}{4}$$
By the Inclusion-Exclusion Principle, the number of ways of distributing $15$ indistinguishable candies to five children so that no child receives more than six candies is
$$\binom{19}{4} - \binom{5}{1}\binom{12}{4} + \binom{5}{2}\binom{5}{4}$$
In how many ways can $15$ indistinguishable candies be distributed to five children if each child receives a different amount of candies.
List the ways $15$ can be expressed as a sum of five distinct nonnegative numbers, starting with
$$0 + 1 + 2 + 3 + 9 = 15$$
and ending with
$$1 + 2 + 3 + 4 + 5 = 15$$
For each of these ways (there are not many), there are $5!$ ways to distribute the candies to the children, depending on which child receives which number of candies.
Best Answer
If each child receives a different number of cakes, then we must distribute at least $0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$ cakes. However, we only have $27$ cakes. Therefore, at least two children must receive the same number of cakes.