Disjoint subsets selection In how many ways can we select $4$ disjoint subsets of the set $\{1,2,\ldots,10\}$?
OK for subsets of 1 element each, there are 10C4 subsets.
For subsets of $2$ elements each, there are $$\binom{10}{2}\cdot\binom{8}{2}\cdot\binom{6}{2}\cdot\binom{4}{2}$$
Then we have subsets of 
$3+3+3+1$: $\binom{10}{3}\cdot\binom{7}{3}\cdot\binom{4}{3}$
Also:
$$
\begin{array}{cc}
1+1+1+2,&1+1+1+3,\\
1+1+2+2,&1+2+2+2,\\
1+1+2+3,&1+2+2+3,\\
2+2+2+3,&1+2+3+3,\\
2+2+3+3,&1+3+3+3,\\
1+1+3+3,&1+2+2+2
\end{array}
$$
Plus the empty subset?
And at the end we multiply all these numbers?  
 A: [2017-10-29]: Answer revised  thanks to OP's comment below.
Note: The problem is somewhat ambiguously formulated. Here we consider an interpretation the author had presumably in mind. 
At first let's have a look at the problem again and check the formulation:


*

*Select four disjoint subsets of $\{1,2,\ldots,10\}$: The empty set $\emptyset$ is a valid subset of $\{1,2,\ldots,10\}$ and since there is no other requirement to the four subsets than being disjoint, we have to consider it as well as any other subset.

*The phrase four disjoint subsets is somewhat unprecise. If we consider four subsets $A_1,A_2,A_3,A_4$ then it could mean either the intersection of all four subsets is empty or it could mean the intersection of each pair of the four subsets is empty. This is not the same. In the first case 
\begin{align*}
A_1=\{1\},A_2=\{1,2\},A_3=\{1\},A_4=\{10\}
\end{align*}
is a valid selection, since $A_1\cap A_2 \cap A_3 \cap A_4=\emptyset$ . In the second case this is not a valid selection, since e.g. $A_1\cap A_2=\{1\}\ne \emptyset$.  Here we consider the second case which is usually meant with this phrase. Observe that in this case   the number of  empty sets  may  vary  between  zero  and  four   within a selection of four subsets.

*Order of subsets: A statement if the order of subsets is  relevant  is missing in the problem. It is not clear for instance if the selections $(A_1,A_2,A_3,A_4)$ and $(A_1,A_4,A_3,A_2)$ should be regarded the same or not. Here we consider the order as relevant. 
In order to avoid ambiguities we clearly state the relevant aspects of the problem:

Problem: We consider four pairwise disjoint subsets of a set $S=\{1,2,\ldots,10\}$ including the empty set where the order of selection of subsets matters.
Solution: We can count the valid selections of four pairwise disjoint subsets $A_1,A_2,A_3,A_4$ surprisingly simple. We start with four empty subsets $A_1,A_2,A_3,A_4$ and  obtain  a valid selection by taking element $1\in S$  and  putting it in one of  four subsets or in no subset at all.This can be done in $5$ different ways. We continue this process for each  of the $10$ elements and conclude:
The  number of valid selections  is $$\color{blue}{5^{10}=9\,765\,625}$$
That's all. 

