How many partitions are there of a 5 element set into 3 parts (of a specific form)?

Question

I am currently trying to understand the number of ways of partitioning a 5 element set into 3 parts. However, I am only interested in partitions with the form $$ \left\{ \{ a, b \}, \{c, d\}, \{e \} \right\} $$ How many ways are there to do this?

My attempt would be to first choose 2 elements from the 5 element set, which can be done in $C(5,2) = 10$ ways.

I would then be required to split the remaining 3 elements in the set into one set of 2 elements and one set of 1 element. To do this, we pick 2 elements from the 3 element set, which can be done in $C(3,2) = 3$ ways.

Thus, by the multiplication principle, the number of ways of splitting the 5 element set into partitions of the desired form is $10 \times 3 = 30$.

However, looking at the solution to this question I have found that the correct answer should have been $$ \mathbf{\frac{1}{2} \times} C(5,2) \times C(3,2) = 15 $$ so my question comes down to this: where has the factor of $\frac{1}{2}$ come from?

Answer 1

Element single: 5 possibilities

4 elements left. 3 options to arrange them

a) (1,2), (3,4)
b) (1,3), (2,4)
c) (1,4), (2,3)

So 5x3=15

The half comes from the fact that the order doesn't matter since they are sets: (1,2) pair is equivalent to the (2,1) one