What things are you given and roughly what knowledge are you supposed to use?
So I am giving you the technical answer. But I cannot explain to you.
The fifth Bell number is .
That is the number of ways to partition a set of five.
But one of those is the set itself, so it does meet the condition of ‘two or more subsets’.
Thus you answer must be .
You cannot have more than 4 members in a subset.
A subset must contain at least one member.
Therefore you could list the possibilities, then count the number of ways those possibilities can occur.
(a) 4 members in a subset and 1 member in the other subset
For this option, you can count using combinations the number of ways to select 4 from 5 or 1 from 5.
(b) (i) 3 members in a subset and 2 members in another subset
You only need to count the number of ways of selecting 3 from 5 or 2 from 5.
(Selecting any 3 automatically selects the remaining 2 and vice versa)
(b) (ii) 3 members in a subset and 2 subsets containing 1 member
You only need to count the number of ways to choose 3 from 5.
(c) (i) 2 members in a subset and 2 members in another subset
..........(the 5th member automatically makes the 3rd subset of 1)
Select 4 of the 5 and count the number of ways to form 2 groups of 2.
Hence, first count the number of ways to select 4 from 5,
then for each group of 4, count the number of ways to pair one member with one of the other 3.
(c) (ii) 2 members in a subset and the other 3 making single-member subsets
Select 2 from 5 as the remainder automatically form the single-member subsets.
(d) One member per subset